Surface group actions and length bounds Brian H. Bowditch Mathematics Institute, University of Warwick, Coventry, CV4 7AL, Great Britain. [September 2010]

0. Introduction.

Let Σ be compact orientable surface, and write Γ = π_{1}(Σ). Let H be a hyperbolic space (in the sense of Gromov [Gr]) admitting a discrete action by Γ. and write M = H/Γ. Let X(Σ) be the set of homotopy classes of essential non-peripheral closed curves in Σ. Let G(Σ) be the curve graph with vertex set V (G) = X(Σ) (i.e. the 1-skeleton of the curve complex [H]). To each γ ∈ X(Σ) we can associate a “stable length” l_{M}^{S}(γ). In [MaM2], the authors define the notion of a “tight geodesic” in G(Σ). The aim of this paper is to describe certain bounds on the stable lengths of curves arising in a tight geodesic. These are a direct generalisation of the “a-priori bounds” described by Minsky in [Mi], where H is hyperbolic 3-space. Here we generalise the approach in [Bo2]. One of the main motivations of this work is its use in describing “coarse models” of hyperbolic 3-manifolds, see [Bo4].

If H is simply connected, we can think of this in terms of the quotient M = H/Γ. Then l_{M}^{S}(γ) is, up to additive constant, equal to the length of a shortest (or nearly shortest) representative of the free homotopy class of γ in M. In fact, there is no essential loss of generality in assuming H to be simply connected, as we will explain later.

To be more precise, we make an assumption on the space H, namely (H1) which corresponds to coarse bounded geometry, and a hypothesis on the action, namely (H2) which substitutes for the Margulis lemma. (It is effectively the conclusion of the Margulis lemma if H happens to be a simply connected manifold of pinched negative curvature.)

Recall that a subset, Q ⊆ H is r-separated if d(x, y) ≥ r for all distinct x, y ∈ Q. We assume:

(H1) (∃s_{0})(∀r ≥ s_{0})(∃n) if Q ⊆ H is r_{0}-separated and diam(Q) ≤ r, then |Q| ≤ n.

(For example, this holds if H is a manifold of bounded Ricci curvature.)

Given x ∈ H and r ≥ 0, write Γ_{r}(x) = {g ∈ Γ | d(x, gx) ≤ r}. We assume:

(H2) There exist r_{0} ≥ s_{0} and ν_{0} such that for all x ∈ H, either |Γ_{r_{0}}(x)| ≤ ν_{0} or ⟨Γ_{r_{0}}(x)⟩ is infinite cyclic.

Clearly (H1) then holds for r_{0} also, so we may as well set s_{0} = r_{0}.

Let Y (Σ) be the set of homotopy classes essential closed curves in Σ. Thus X(Σ) ⊆ Y (Σ). We write Y (∂Σ) ⊆ Y (Σ) for the (possibly empty) set of boundary components. Each γ ∈ Y (Σ) determines an element g ∈ Γ defined up to conjugacy. Its stable length is defined as ||g|| = lim_{n→∞} \frac{1}{n}d(x, g^{n}x) for some, hence any, x ∈ H. We set l_{M}^{S}(γ) = ||g||. We will l_{M}^{S}(∂Σ) = ∑{l_{M}^{S}(γ) | γ ∈ Y (∂Σ)}. (This is 0 if ∂Σ = ∅.)

It will be convenient to view the following as another hypothesis on the action. We fix some L ≥ 0 and suppose:

Surface group actions and length bounds

(H3) l_{M}^{S}(∂Σ) ≤ L.

Definition : By the parameters of the action we mean collectively the hyperbolicity constant of H, the topological type of Σ, the constant r_{0} and function [r ↦ n(r)] featuring in (H1), the constants r_{0} and ν_{0} featuring in (H2) and the constant L featuring in (H3).

(Recall that we have set s_{0} = r_{0} for convenience of notation.)

Given l ≥ 0, let Y (M, l) = {γ ∈ Y | l_{M}^{S}(γ) ≤ l}. We write X(M, l) = X(Σ) ∩ Y (M, l). The notion of a “tight geodesic” we use here is that of [Bo2] which is a slight gener-

alisation of the original in [MaM]. See Section 7 for further discussion.

Theorem 0.1 : Suppose Γ = π_{1}(Σ) acts on a hyperbolic space H satisfying (H1), (H2), (H3) above. Suppose that γ_{0}γ_{1} ⋯ γ_{n} is a tight geodesic in G(Σ) with γ_{0}, γ_{p} ∈ X(M, l). Then γ_{i} ∈ X(M, l′) for all i ∈ {0, …, n}, where l′ depends on l and the parameters of the action.

If M ≅ int(Σ) × R is a complete hyperbolic 3-manifold, then l_{M}^{S} is just the length of the corresponding closed geodesic, γ_{M}, in M (or 0 if this is parabolic). Thus, in this case, Theorem 0.1 is just a variant of Minsky’s a-priori bounds theorem [Mi]. The formulation we give here ties in with that in [Bo2] (cf. Theorem 1.3 of that paper.)

Theorem 0.1 divides into two cases, namely when Σ is exceptional (that is, a oneholed torus or four-holed sphere) or non-exceptional. The proofs are quite different, and the latter occupies most of this paper. The former is largely independent and described in Section 8. In this case, G(Σ) is a Farey graph, and every geodesic is deemed tight.

In the course of the proof, we show:

Theorem 0.2 : Suppose Γ = π_{1}(Σ) acts on a hyperbolic space, H satisfying (H1), (H2) and (H3). Then there exist l_{0}, h_{0} ≥ 0, depending only on the parameters of the action such that X(M, l_{0}) is non-empty and h_{0}-quasiconvex in G(Σ). Moreover, if l ≥ l_{0}, then there is some h_{1} ≥ 0 depending only in l and the parameters of the action such that X(M, l) ⊆ N(X(M, l_{0}), h_{1}).

Here N(., r) denotes the r-neighbourhood.

The following variant of Theorem 0.1 will also prove useful in applications (cf. Theorem 1.4 of [Bo2].)

Theorem 0.3 : Suppose Γ acts on H satisfying (H1), (H2), (H3). There is some r_{0} depending only or the parameters of the action such that γ_{0}, …, γ_{p} is a tight geodesic in G(Σ) and r + r_{0} ≤ i ≤ p − r − r_{0}, where r = max{d(γ_{0}, X(M, l)), d(γ_{p}, X(M, l))}, then γ_{i} ∈ X(M, l′), where l′ depends only on the parameters of the action and l.

(Note we can always take l ≥ l_{0}, so that X(M, l_{0}) ≠ ∅.)

In combination with the above, this gives “a-priori bounds” for “hierarchies” in G(Σ) in the same manner as described in [Bo2]. Such bounds for 3-manifolds were originally described in [Mi], and were central to the proof of the Ending Lamination Conjecture in [Mi,BrCM]. The application to hierarchies will be discussed in Section 9.

In order to simplify the exposition, we will make a few additional assumptions on M, namely:

(∗) H is a simply connected geodesic space, and Γ acts freely on H. Moreover, for each g ∈ Γ, the minimum |g| = min{d_{H}(x, gx) | x ∈ H} is attained.

We can do this without any loss of generality as far as the main results are concerned. It is sufficient to note that any hyperbolic space admitting any Γ-action is equivariantly quasi-isometric to one satisfying (∗). (This is described in [Bo3] using a variation on the Rips complex construction.) Moreover the hypotheses and conclusions are essentially quasiisometry invariant. Alternatively one could readily reinterpret the arguments presented here without using (∗), though they become more clumsy to formulate.

Assuming (∗), we see that π_{1}(M) ≡ Γ, and we can associate to each γ ∈ Y (Σ) a free homotopy class of curves in M. Such a homotopy class admits a minimal length representative, γ_{M}, (not necessarily unique). We write l_{M}(γ) = length(γ_{M}) = |g|, where g ∈ Γ is corresponding element (or conjugacy class) in Γ.

We note that l_{M}^{S}(γ) ≤ l_{M}(γ), and that l_{M}(γ) − l_{M}^{S}(γ) is bounded above by some fixed multiple of the hyperbolicity constant. Thus, in the above theorem we could equally well replace l_{M}^{S} by l_{M}.

Definition : We refer to a space M arising in the manner above (for an action of Γ = π_{1}(Σ) on a hyperbolic space H satisfying (H1), (H2), (H3) and (∗) as a coarse hyperbolic manifold We refer to the parameters of the action also as the parameters of M.

The proofs of the main results follow a similar strategy to that in [Bo2]. This was originally inspired by an idea of Bestvina and Fujiwara, which in turn was inspired by an argument of Luo. The basic idea is as follows.

Suppose we have a tight geodesic, (γ_{i})_{i=0}^{p} in G(Σ) whose elements are realised as shortest curves (γ_{i})_{M}, in M. Suppose that the realisations of γ_{0} and γ_{p} have bounded length, but those of some intermediate γ_{i} are very long. There is a sense in which these γ_{i} “fill out” certain subsurfaces of Σ. Tightness implies that consecutive surfaces overlap is such a way that we can find curves in them that shortcut the geodesic (γ_{i})_{i}, thereby giving a contradiction. In [Bo2], these subsurfaces were described using pleating surfaces. Here, we will use a coarse analogue of that construction.

This construction is motivated by the following observation. Suppose that σ is a (constant curvature) hyperbolic metric on Σ, and that γ is a simple closed curve realised as a closed geodesic, γ_{σ}, in (Σ, σ). One can give a topological description of a metric neighbourhood, N, of γ_{σ} in (Σ, σ) along the following lines. Suppose that ϵ is an arc in Σ meeting γ exactly at its endpoints — that is a “bridge arc”. If ϵ can be realised as a short arc in (Σ, σ), then at each of its endpoints there are long arcs in γ_{σ} which follow each either within a bounded distance. Properly quantified, the neighbourhood, N, is determined by the set of such arcs, and can thus be described without explicit reference to the small scale structure of (Σ, σ). We only need to have things defined up to a bounded distance.

We can perform an analogous construction with realisations of curves in M, instead. Arcs in (γ_{i})_{M} which remain close over a large distance determine a certain class of bridge arcs, which in turn determine a subsurface of Σ, defined up to homotopy in Σ. If γ_{i} and γ_{i+1} are consecutive elements of a tight geodesics whose realisations in M are sufficently long, then these subsurfaces overlap. This gives rise to a sequence of curves that shortcut the geodesic (γ_{i})_{i}.

Much of the work in this paper is involved in describing this process more formally, and checking that it gives rise to something sensible. In [Bo2] this used pleating surfaces, and the uniform injectivity theorem. Here we will need a coarse analogue of this (see Section 6). This in turn uses the fact, from [Bo3], that simple closed curves realised in M follow close to a train track in M.

The above discussion applies to the case where Σ is “non-execptional”. The exceptional cases of the one-holed torus and four-holed sphere will be dealt with by in independent argument in Section 8.

1. The thick-thin decomposition.

We describe how (H1) and (H2) give rise to a thick-thin decomposition of M analogous to that of a hyperbolic manifold.

If γ is a closed curve Σ, we write γ̃ for the entire preimage of γ in Σ̃. We will use γ̆ for a component of γ̃, that is a bi-infinite arc. This is stabilised by some infinite cyclic subgroup, G(γ̆) = ⟨g⟩ ⊆ Γ, where g represents (the conjugacy class of) γ in Γ = π_{1}(Σ). Note that every cyclic subgroup of Γ has this form.

We write Y_{0}(Σ) ⊆ Y (Σ) for the set of primitive closed curves (so that X(Σ) ⊆ Y_{0}(Σ)) and write Ỹ_{0}(Σ) ⊆ Ỹ (Σ) for the set of lifts of these to Σ̃. Note that γ̆ ∈ Ỹ_{0} if and only if G(γ̆) is a maximal cyclic subgroup of Γ.

Now suppose Γ acts on hyperbolic space H satisfying (H1) and (H2) as described in Section 0. In particular, π_{1}(M) ≡ Γ, where M = H/Γ. We describe the thin part of M.

Recall that Γ_{r}(x) = {g ∈ Γ | d(x, gx) ≤ r}. Given r ≥ 0, ν ∈ N and G ≤ Γ, we set Δ̃(G; r, ν) = {x ∈ H | |G ∩ Γ_{r}(x)| ≥ ν}. We abbreviate Δ̃(r, ν) = Δ̃(Γ; r, ν). Thus Δ̃(r, ν) is Γ-invariant and we set Δ(r, ν) = Δ̃(r, ν)/Γ ⊆ M.

If G ≤ Γ is any subgroup, then for all r, ν, we have N(Δ̃(G; r, ν), s) ⊆ Δ̃(G; r + 2s, ν). This follows directly from the triangle inequalities. Moreover, applying (H1), we see that if r ≥ r_{0} and ν ∈ N there is some ν′ ∈ N such that for any G, Δ̃(G; r, ν′) ⊆ Δ̃(G; r_{0}, ν). Putting these together, we obtain:

Lemma 1.1 : For all r ≥ r_{0}, ν ∈ N and s ≥ 0, then there is some ν′ ∈ N such that for all G ≤ Γ we have N(Δ̃(G; r, ν′), s) ⊆ Δ̃(G; r_{0}, ν). ♢

Now given any γ̆ ∈ Ỹ_{0}(Σ), we set Δ̃(γ̆; r, ν) = Δ̃(G(γ̆); r, ν). Given any γ ∈ Y_{0}, we set Δ(γ̆; r, ν) for the image of Δ̃(G; r, ν) in M, where γ̆ is any lift of γ.

Lemma 1.2 : If ν ≥ ν_{0}, then Δ̃(r_{0}, ν) is a disjoint union of Δ̃(γ̆; r_{0}, ν) as γ̆ ranges over Ỹ.

Note, in particular, that this implies that Δ(r_{0}, ν) = ⨆_{γ∈Y_{0}} Δ(γ; r_{0}, ν).

Proof : Clearly ⋃_{γ̆∈Ỹ_{0}} Δ̃(γ̆; r_{0}, ν) ⊆ Δ̃(r_{0}, ν). For the reverse inclusion, suppose x ∈ Δ(r_{0}, ν). Thus |Γ_{r_{0}}(x)| ≥ ν ≥ ν_{0}, and so, by (H2), ⟨Γ_{r_{0}}(x)⟩ is infinite cyclic, and hence contained in a maximal infinite cyclic subgroup, G(γ̆) for some γ̆ ∈ Ỹ_{0}. Thus x ∈ Δ̃(γ̆; r_{0}, ν) as required.

For disjointness, suppose γ̆, δ̆ ∈ Ỹ_{0}, with x ∈ Δ̃(γ̆; r_{0}, ν) ∩ Δ̃(δ̆; r_{0}, ν). Now ⟨Γ_{r_{0}}(x)⟩ is contained in a maximal infinite cyclic subgroup G ≤ Γ and meets both G(γ̆) and G(δ̆) non-trivially. It follows that G(γ̆) = G(δ̆) = G, and so γ̆ = δ̆. ♢

We shall abbreviate Δ_{0} = Δ(r_{0}, ν_{0}), Δ̃_{0} = Δ̃(r_{0}, ν_{0}), Δ_{0}(γ) = Δ(γ; r_{0}, ν_{0}) and Δ̃_{0}(γ̆) = Δ̃_{0}(γ̆; r_{0}, ν_{0}).

Note that if ν ≥ ν_{0} and γ ∈ Y_{0}, then Δ(γ; r_{0}, ν) = Δ_{0}(γ) ∩ Δ(r_{0}, ν) and Δ̃(γ̆; r_{0}, ν) = Δ_{0}(γ̆) ∩ Δ̃(r_{0}, ν).

We remark that one can show that the sets Δ̃(γ̆; r_{0}, ν) are uniformly quasiconvex in M̃, though we will not be explicitly using that in this paper.

2. The track construction.

We recall some consequences of the track construction described in [Bo3].

We fix some ν_{0} as given by (H1), (H2), and let Δ_{0} ⊆ M be the corresponding thin part as described in Section 1.

Now let γ ⊆ Σ be a multicurve, and let q : γ ⟶ M be a realisation, so that q(γ) has minimal length in its homotopy class. We write d_{γ} for the induced path-metric on γ. We say that a component, α, of γ is short if its length, l_{M}(α) = length(q(α)) is less than some fixed constant, which for convenience we can set to be the hyperbolicity constant of M̃. This means that if α is not short, then each lift q̃ : ᾰ ⟶ M̃ is uniformly quasigeodesic (with constants depending only on the hyperbolicity constant).

By a multitrack we mean a train track, τ, in the sense of Thurston, though dropping the condition that it be connected. In other words, it is a graph embdded in the interior of Σ, such that there are no nullgons, monogons, digons or smooth annuli in the complement. (For terminology, see for example, [PH].)

We recall the following definitions from [Bo3]. By a loop, δ ⊆ τ mean a circular subtrack conisisting of a single branch and a single switch of τ. It is simple if at the switch, there is exactly one incident half-branch on each side of δ, and arranged so that there is a local trainpath of τ meeting δ precisely at the switch. If δ happens to be a boundary component, then there is just one branch emerging from the switch, and we drop the final condition. Note that distinct simple loops are necessarily disjoint. An annular neighbourhood, A, of δ is simple if A∩τ consists of either just δ, or else γ together with an interval in each of the adjacent edges. By a special set of simple loops, we mean a preferred (possibly empty) set, S, of loops, together with a set, A(S) = {A(δ) | δ ∈ S}, of disjoint simple annular neighbourhoods.

Now the construction of [Bo3] gives us a multitrack, τ ⊆ Σ, a path-metric d_{τ} on τ, a carrying map p : γ ⟶ τ, a map f : τ ⟶ M in the correct homotopy class, a (possibly empty) collection, S_{0} = S, of simple loops in τ and associated simple annuli, A_{0}(δ) = A_{0}(δ, τ) for δ ∈ S_{0}, satifying various geometric conditions. Writing A_{0}(Σ) = ⋃_{δ∈S_{0}} A_{0}(δ), we have:

(A1) p : (γ, d_{γ}) ⟶ (δ, d_{τ}) is ξ_{0}-lipschitz.

(A2) For all x ∈ γ̃, d(q̃(x), f̃p̃(x)) ≤ K_{1}. (A3) τ \ A_{0}(Σ) has length at most K_{2}. (A4) f|τ \ ⋃ S_{0} ⟶ (M, d) is ξ_{1}-lipschitz. (A5) For each δ ∈ S_{0}, f(A_{0}(δ)) ⊆ Δ_{0}(δ).

Here all the constants depend only on the parameters of M.

For suitably chosen ν_{2} ≥ ν_{1} ≥ ν_{0} we set Δ_{1} = Δ(r_{0}, ν_{1}) and Δ_{2} = Δ(r_{0}, ν_{2}) so that Δ_{2} ⊆ Δ_{1} ⊆ Δ_{0}.

In addition, we can find S_{2} ⊆ S_{1} ⊆ S_{0}, and special annuli A_{1}(δ) = A_{1}(δ, τ) ⊆ A_{0}(δ) for each δ ∈ S_{1}, and A_{2}(δ) = A_{2}(δ, τ) ⊆ A_{1}(δ) for each δ ∈ S_{2}, with the following properties:

(A6) τ \ A_{2}(Σ) has length at most K′_{2}. (A7) f|τ \ ⋃ S_{2} ⟶ M is ξ′_{1}-lipschitz. (A8) For each δ ∈ S_{2}, f(A_{2}(δ)) ⊆ Δ_{2}(δ).

(A9) For each δ̆ ∈ S̃_{1}, f^{−1}(Δ_{1}(δ̆)) ⊆ A_{1}(δ̆).

Here the new constants depend also on the choice of ν_{1} and ν_{2}.

In fact, we can arrange that N(Δ_{2}, s) ⊆ Δ_{1} and N(Δ_{1}, s_{0}) ⊆ Δ_{0}, where s_{0}, s ≥ 0, are chosen sufficiently large as described later. (In practice, we will be using two different choices of Δ_{1} and Δ_{2}, in Sections 3 and 7 respectively. Ultimately, however, these choices will depend only the parameters of M.)

Note that a consequence of (A5) is that if δ̆ ∈ S̃_{0}, then f̃(A_{0}(δ)) ⊆ Δ̃_{0}(δ̆).

The above statements, (A1)–(A8), are all consequences of the corresponding statements, (T1)–(T8), listed in [Bo3]. (We set K − 1 = h_{0}, K_{2} = h_{1}(ν_{0}), ξ_{1} = ξ_{1}(ν_{1}), ξ′_{1} = ξ_{1}(ν_{2}) in the notation of that paper. Note that (A5) and (A8) are slightly weaker than the corresponding statements of [Bo3].) Indeed, for these we could take any ν_{2} ≥ ν_{0}. (In that paper, τ was a track filling Σ, and we were taking ∂Σ ⊆ τ. To get a multitrack in the sense defined above, we can simply replace τ by the subtrack, p(γ), pushed into the interior of Σ, if necessary, so that τ ∩ ∂Σ = ∅. Property (A9) is specific to the set up in this paper, and calls for further comment.

We can suppose that S_{0} ≠ ∅, otherwise nothing remains to be proven. We set S_{1} = S_{0} and A_{1}(δ) = A_{0}(δ). Let s_{0} ≥ ξ_{0}K_{2}. We suppose that ν_{1} ≥ ν_{0} is large enough so that N(Δ_{1}, s_{0}) ⊆ Δ_{0}. Let δ̆ ∈ S̃_{1}. We claim that f^{−1}(Δ_{1}(δ̆)) ⊆ A_{0}(δ̆). Suppose that x ∈ τ̃, with f(x) ∈ Δ_{1}(δ̆), and that x ∉ A_{0}(δ̆). Then we can connect x to some point y ∈ A_{0}(ϵ̆) by a path in τ̃ of d_{τ} -length at most K_{2}, where ϵ̆ ∈ S̃_{0} \ {δ̆}. Thus d(f̃(x), f̃(y)) ≤ ξ_{0}K_{2} ≤ s_{0}.

But since f̃(A_{0}(ϵ)) ⊆ Δ̃_{0}(ϵ̆), it now follows that ϵ̆ = δ̆ giving a contradiction. This now justifies the statement (A9).

Suppose that γ has no short components, and that γ̆ is a component of γ̃ ⊆ Σ̃. Now p̃ : γ̆ ⟶ τ̃ is injective, and p̃(γ̆) is a bi-infinite trainpath. Also f̃p̃ and τ are a bounded distance apart. Now f̃ and p̃ are uniformly lipschitz, and q̃ is uniformly quasigeodesic, so it follows that both p̃: γ̆ ⟶ τ̃ and f̃|p̃(γ̆) ⟶ M̃ are both uniformly quasigeodesic.

We also want to relate this to combinatorial distance in τ, where each branch of τ has unit length. Suppose that ζ̆ ⊆ γ̆ is an arc such that fp(ζ) ∩ Δ_{2} = ∅. In particular, this means that p(ζ) ⊆ τ \ A_{2}(δ). Since each component of τ \ A_{2}(Σ) has bounded diameter, there is a linear bound on the length of p̃(ζ̆) in terms of the number of branches it passes through. Since p̃: ζ̆ ⟶ τ̃ is quasigeodesic, this also places a linear bound on the length of ζ̆.

Conversely, if ζ is an arc in γ of bounded length, with fp(ζ) ∩ Δ_{2} = ∅, then there is a bound on the number of branches of τ that pζ can pass through, since each time pζ passes through a given branch, we get an element of bounded displacement in M̃, and the number of these is bounded by the assumption that fp(ζ) ∩ Δ_{2} = ∅. Thus, if ζ is any path with fp(ζ) ∩ Δ_{2} = ∅, then the combinatorial length of pζ is linearly bounded above by the d_{γ}-length of ζ.

3. The uniform injectivity theorem.

In this section, we prove a result will serve as a substitute for the uniform injectivity theorem for pleated surfaces in 3-manifolds. The argument we give here is based on a construction of Namazi and Souto [NS], though some reintepretation is necessary to deal with coarse hyperbolic spaces. I am grateful to Hossein Namazi for explaining the main idea of their proof to me.

Suppose γ is a multicurve with no short components (i.e. all components have length at least the hyperbolicity constant in M). Let q : γ ⟶ M be a realisation, and let q̃: γ̃ ⟶ M̃ be its lift to the preimage, γ̃, to Σ̃. We have observed that each component is uniformly quasigeodesic.

Suppose that ζ ⊆ γ is any arc, and ζ̆ ⊆ γ̃ is a lift. We can always find a homeomorphism, λ : I ⟶ ζ, where I ⊆ R is some interval, such that for all t, u ∈ I we have |d(q̃λ̃(t), q̃λ̃(u)) − |t − u|| bounded above in terms of the hyperbolicity constant of M̃. We refer to such a homeomorphism as an ambient parameterisation of ζ. (The appropriate bound chosen will be apparent in what follows.)

Suppose that ζ̆, ζ̆′ ⊆ γ̃ are two oriented arcs with positive and negative endpoints, x, y and x′, y′ respectively. We write dM̃(q̃ζ̆, q̃ζ̆′) = max{dM̃(q̃x, q̃x′), dM̃(q̃y, q̃y′)}.

We note the following:

Lemma 3.1 : There is some h_{0}, depending only on the hyperbolicity constant with the following property. Suppose that ζ̆, ζ̆′ are as above, and let h = dM̃(q̃ζ̆, q̃ζ̆′). Then there are intervals, I, J, J′ ⊆ R, with I ⊆ J ∩ J′, with each component of J \ I and of J′ \ I having length at most h, and positively oriented ambient parameterisations λ : J ⟶ ζ and λ′ : J′ ⟶ ζ′ such that for all t ∈ I, we have dM̃(q̃λ̃(t), q̃λ̃′(t)) ≤ h_{0}. ♢

This is just an expression of the “fellow travelling” property of uniform quasigeodesics in a hyperbolic space: if two quasigeodesics are a bounded distance apart at their endpoints, then they remain a uniformly bounded distance apart over most of their length.

Given maps, v, v′ : I ⟶ M̃, we write v ∼_{h} v′ to mean that dM̃(v(t), v′(t)) ≤ h for all t ∈ I. We will abbreviate ∼_{h_{0}} to ∼, where h_{0} is the constant of Lemma 3.1.

Definition : Suppose ζ̆, ζ̆′ ⊆ γ̃ are oriented arcs. We say that q̃ζ̆ and q̃ζ̆′ are parallel if they admit positively oriented ambient parameterisations λ : I ⟶ ζ and λ′ : I′ ⟶ ζ′ such that q̃λ̃ ∼ q̃λ̃′.

Thus, modulo reducing the lengths of I and I′ by at most h_{0}, this is equivalent to saying that dM̃(q̃ζ̆, q̃′ζ̆′) ≤ h_{0}.

Recall that in Section 2 we defined three nested thin parts of M, namely Δ_{2} ⊆ Δ_{1} ⊆ Δ_{0}, with N(Δ_{1}, s_{0}) ⊆ Δ_{0} and N(Δ_{2}, s) ⊆ Δ_{1}. We will postpone the explicit specification of s_{0} and s. We can certainly assume that s is sufficiently large so that there is some ν with ν_{1} ≤ ν ≤ ν_{2}, giving an intermediate thin part, Δ_{2} ⊆ Δ_{ν} ⊆ Δ_{1}, and with N(Δ_{2}, s_{2}) ⊆ Δ_{ν} and N(Δ_{ν}, s_{1}) ⊆ Δ_{1}. We will specify required bounds on s_{1} and s_{2}, depending on the parameters of M, later.

Now let τ be an associated multitrack as discussed in Section 2. With s_{1} is sufficiently large, we have the following:

Lemma 3.2 : q(γ) ∩ Δ_{ν} ⊆ ⋃_{δ∈S_{1}} Δ_{ν}(δ).

Proof : We know that Δ_{ν} is a disjoint union of Δ_{ν}(ϵ) as ϵ varies over Y_{0} = Y_{0}(Σ). Let x ∈ γ with q(x) ∈ Δ. Thus q(x) ∈ Δ_{ν}(ϵ) for some ϵ ∈ Y_{0}. Now d(q(x), fp(x)) ≤ K_{3} and so fp(x) ∈ N(Δ_{ν}(ϵ), K_{3}) ⊆ Δ_{1}(ϵ) (provided s_{1} ≥ K_{3}). By the property of Δ_{1} discussed in Section 2, we have p(x) ∈ f^{−1}(Δ_{1}) ⊆ A_{1}(Σ). That is, p(x) ∈ A_{1}(δ) for some δ ∈ S_{1}, and so fp(x) ∈ Δ_{0}(δ) (again from Section 2). But we already have fp(x) ∈ Δ_{1}(ϵ) ⊆ Δ_{0}(ϵ) and so Δ_{0}(δ) ∩ Δ_{0}(ϵ) ≠ ∅. Thus ϵ = δ. ♢

The idea behind the uniform injectivity theorem is that if two intervals in γ follow each other in M then they also follow each other in Σ. We shall express this in terms of the associated multitrack, τ, described in Section 2. In certain cases, it can be reformulated in terms of a hyperbolic structure on Σ as we describe in Section 5.

Definition : We say that two oriented trainpaths, ζ̆, ζ̆′ in τ̃ intersect positively if ζ̆ ∩ ζ̆′ is a non-trivial interval, and ζ̆ and ζ̆′ are oriented consistently on this interval.

(Note that the intersection of any two trainpaths in τ̃ is connected.)

If π̃, π̃′ : I ⟶ π̃ are parameterisations of ζ̆ and ζ̆′, this can be expressed by saying that there exist t < u ∈ I and t′ < u′ ∈ I with π̃(t) = π̃′(t′) and π̃(u) = π̃′(u′).

In the following formulation of the uniform injectivity theorem, Δ_{ν} ⊆ M is the thin part introduced above. For the proof, we can assume without loss of generality, that ν is sufficiently large that it has the properties given above.

Theorem 3.3 : Given ν > 0, there is some constant, T_{0} > 0, depending only on the parameters of M with the following property. Suppose that γ is a multicurve with no short components. Let q : γ ⟶ M be a realisation, and let τ be an associated multitrack with carrying map, p : γ ⟶ τ. Suppose that ζ, ζ′ are arcs in γ of d_{γ}-length at least T_{0} with (qζ ∪ qζ′) ∩ Δ_{ν} = ∅. Let ζ̆, ζ̆′ be lifts to γ̃. If q̃ζ̆ and q̃ζ̆′ are parallel, then p̃ζ̆ and p̃ζ̆′ intersect positively.

Proof : From the earlier discussion, we can assume that ζ = λ(I), ζ′ = λ′(I), where I = [−v_{0}, v_{0}], and λ, λ′ : I ⟶ γ̆, are ambient parameterisations, and that q̃λ̃ ∼ q̃λ̃′.

Recall that Δ_{2} is chosen so that N(Δ_{2}, s_{2}) ⊆ Δ_{ν} and that f(τ ∩ A_{2}(Σ)) ⊆ Δ_{2} (by the construction in Section 2). For all t ∈ I, d(qλ(t), fpλ(t)) ≤ K_{2} ≤ s_{2} provided we assume that s_{2} ≥ K_{2}. It follows that fpλ(I) ∩ Δ_{2} = ∅, and so pλ(I) ∩ A_{2}(Σ) = ∅. Similarly, pλ′(I) ∩ A_{2}(Σ) = ∅.

Now τ \ A_{2}(Σ) has bounded length, an wo we can cover it by a set, I, of intervals of bounded length, each lying in some branch of τ, and with |I| bounded by some number, N, depending only on ν andthe parameters of M. From the above discussion, we have pλ(I) ∩ pλ′(I) ⊆ ⋃ I. We write Ĩ for the set of lifts of these intervals to τ̃.

Now write π̃ = p̃λ̃ : I ⟶ τ̃ and π̃′ = p̃λ̃′ : I ⟶ τ̃, and write π and π′ for their projections to M. We write π_{+} and π_{−} for the trainpaths π|[0, v_{0}] and π|[−v_{0}, 0] respectively. We fix some u_{0} ≥ 0, to be determined later. We consider the pairs (π(t), π′(t)) as t varies from 0 to v_{0}. Each such pair lies in θ × θ′ for some θ, θ′ ∈ I. There are at most N^{2} possibilities. Thus, if v_{0} is sufficiently large, we find some fixed θ, θ′ ∈ I and t_{i} with 0 < t_{0} < t_{1} < ⋯ < t_{m} < v_{0} with t_{i} + u_{0} < t_{i+1} for all i, such that π(t_{i}) and π′(t_{i}) lie in the interiors of θ and θ′ respectively, such that π and π′ pass through these intervals in the same direction at these points, and where m ∈ N is some fixed constant depending only on the parameters, as determined below. Now if u_{0} is sufficiently large, we see that for each i, π must leave the interval θ, and π′ must leave the inerval θ′, between t_{i} and t_{i+1}.

Now if i < j, we can represent π|[0, t_{j}] as a union of a path α_{i} = π|[0, t_{i}] and a path β_{ij} = π|[t_{i}, t_{j}] whose image is a closed trainpath in τ (at least if we allow ourselves to slide t_{i} along the interval θ). This represents an element g_{ij} ∈ π_{1}(Σ, π(0)). Writing g_{i} = g_{0i}, we see that g_{ij} = g_{j}g_{i}^{−1}. Similarly, we can write π′|[0, t_{j}] = α′_{i} ∪ β′_{ij}. This is represented by g′_{j}(g′_{i})^{−1} ∈ π_{1}(Σ, π′(0)). We can interpret this in terms of τ̃ ⊆ Σ̃ as follows.

Let π̃(t_{i}) ∈ θ̃_{i} ∈ Ĩ and π̃′(t_{i}) ∈ θ̃′_{i} ∈ Ĩ. We connect π̃(0) to π̃′(0) by a path ϵ̃. This projects to a path ϵ in Σ from π(0) to π′(0), thereby allowing us to identify π_{1}(Σ, π(0)) ≡ π_{1}(Σ, π′(0)) ≡ Γ. Moreover, we see that θ̃_{i} = g_{i}θ̃_{0} and θ̃′_{i} = g′_{i}θ̃′_{0}.

Let x̃_{i} = q̃λ̃(t_{i}) ∈ M̃ and ỹ_{i} = f̃p̃λ̃(t_{i}) ∈ M̃. Thus, dM̃(x̃_{i}, ỹ_{i}) ≤ K_{3}, by the property of τ described in Section 2. We similarly define x̃′_{i} and ỹ′_{i}, and again have dM̃(x̃′_{i}, ỹ′_{i}) ≤ K_{3}. Moreover, since q̃λ̃ ∼ q̃λ̃′, we have d(x̃_{i}, x̃′_{i}) ≤ h_{0} for all i.

Now π̃(t_{i}), g_{i}π̃(t_{0}) ∈ θ̃_{i}, and so dτ̃(π̃, g_{i}π̃(t_{0}) is bounded since f̃ is lipschitz, and therefore is dM̃(ỹ_{i}, g_{i}ỹ_{0}) = dM̃(f̃π̃(t_{i}), g_{i}f̃π̃(t_{0})) is bounded. It follows that dM̃(x̃_{i}, g_{i}x̃_{0}) is bounded. Similarly, dM̃(x̃′_{i}, g′_{i}x̃′_{0}) is bounded. This therefore also bounds dM̃(x̃′_{0}, g_{i}^{−1}g′_{i}x̃′_{0}) = dM̃(g_{i}x̃′_{0}, g′_{i}x̃′_{0}) ≤ dM̃(x̃_{0}, x̃′_{0}) + dM̃(g_{i}x̃_{0}, g′_{i}x̃′_{0}). That is, for all i, g_{i}^{−1}g′_{i} ∈ Γ_{r}(x̃_{0}), where r depends only on the parameters of the action. But x̃_{0} ∈ q̃λ̃(I) = q̃ζ̆ and so, by assumption, x_{0} ∉ Δ_{ν}. Thus, by (M2), there is some bound, m, on the cardinality of Γ_{r}(x̃_{0}), again depending only on the parameters of the action. This retrospectively gives us our constant, m, introduced above. It now follows that there exist i < j so that g_{i}^{−1}g′_{i} = g_{j}^{−1}g′_{j} and so g_{j}g_{i}^{−1} = g′_{j}(g′_{i})^{−1}.

Now let α_{+} = α_{i}, β_{+} = β_{ij}, α_{+} = α′_{i} and β′_{+} = β′_{ij} be as above. Note that σ_{+} = α_{+} ∪ β_{+} and σ′_{+} = α′_{+} ∪ β′_{+} represent the same element of Γ. In particular, β_{+} and β′_{+} are freely homotopic in Σ, and so represent the same closed trainpath in τ. Moreover, since π̃, π̃′ : I ⟶ τ̃ are uniform quasigeodesics, we can arrange that this closed trainpath has τ-length at least a given fixed constant (to be chosen below) by choosing the constant u_{0} sufficiently large.

We now carry out the same argument with π_{−} and π′_{−}. We get paths α_{−} ∪ β_{−} and α′_{−} ∪ β′_{−} representing the same element of Γ, and so β_{−} and β′_{−} are the same closed trainpath in τ. We write σ = σ_{−} ∪ σ_{+} and σ′ = σ′_{−} ∪ σ′_{+}. Let σ̆ and σ̆′ be lifts of these arcs to τ̃.

Back in τ̃, we have two bi-infinite paths β̆_{+} = β̆′_{+} and β̆_{−} = β̆′_{−} such that the lifted trainpaths σ̆ and σ̆′ both run from β̆_{−} in the same sense and into β̆_{+} in the same sense.

We distinguish two possibilities as follows.

Case (1): β̆_{−} ∩ β̆_{+} = ∅.

In this case σ̆ and σ̆′ both cross from β̆_{−} to β̆_{+} along the same trainpath and so, in particular, intersect positively. (We use the fact that a train track has no complementary digons.)

Case (2): β̆_{−} ∩ β̆_{+} ≠ ∅.

For this we will use the following general principle. Suppose that μ, μ′ are overlapping trainpaths in τ̃, with μ ∪ μ′ a trainpath. Suppose that f̃μ and f̃μ′ are quasigeodesic in M̃. Then if the dτ̃-length of μ∩μ′ is sufficiently large in relation to the quasigeodesic constants, then f̃(μ ∪ μ′) is also quasigeodesic, with constants depending only on the original. This follows from the fact that being quasigeodesic in a Gromov hyperbolic space is a local property. In fact, the above extends to any sequence of intervals, each overlapping the next in a sufficiently long arc.

Now we know that f̃σ̆ and f̃σ̆′ are uniformly quasigeodesic in M̃ (since they follow, respectively, within a bounded distance of q̃ζ̆ and q̃ζ̆′). We can therefore choose some l sufficiently large for the above construction to work, with these quasigeodesic constants, and choose u_{0} in the above construction, large enough so that β and β′ both have length at least l.

Now let ω be the trainpath β̆_{−} ∩ β̆_{+}. (This will be all of β̆_{−} and β̆_{+} if these happen to be equal.) Now since σ and σ′ both wrap around β_{−} and β_{+} in the same direction, it follows that if σ̆ and σ̆′ intersect, then they must intersect positively. Thus, we assume, for contradiction that σ̆ ∩ σ̆′ = ∅. Since both cross from β̆_{−} to β̆_{+}, we see that, up to interchanging σ and σ′, we have σ̆ ⊆ β̆_{−} and σ̆′ ⊆ β̆_{+}, and that σ̆ ∪ ω ∪ σ̆′ is a trainpath. We can also assume that β_{−} is no longer than β̆_{−} (by interchanging the positive and negative directions).

Now σ must wrap at least twice around β̆_{−}, and so from the above principle, we see that if length(β_{−}) ≥ l is sufficiently large, the f̃β̆_{−} will be uniformly quasigeodesic (since β̆_{−} is made up of a sequence of overlapping images of σ̆). In particular, f̃ω is uniformly quasigeodesic.

But now σ also wraps at least once around β_{−}, and so it follows that length(σ ∩ ω) ≥ length(β_{+}) ≥ l. Similarly, length(σ̆′ ∩ ω) ≥ length(β_{−}) ≥ l. So again, since f̃σ̆, f̃σ̆′ and f̃ω are all quasigeodesic, it follows that f̃(σ̆ ∪ ω ∪ σ̆′) is uniformly quasigeodesic. But now, from the original construction, f̃σ̆ and f̃σ̆′ remain a bounded distance apart in M̃. Thus, again if l is sufficiently large, we get a contradiction.

We have shown that, in both cases, σ̆ and σ̆′, and hence p̃ζ̆ and p̃ζ̆′ intersect positively. This involved choosing v_{0} sufficiently large in relation to the parameters of M. This is allowed, since T_{0} is linearly bounded above in terms of v_{0}, so it is enough to choose T_{0} sufficiently large. ♢

An immediate consequence of Theorem 3.3 is the following.

Suppose that I ⊆ R is an interval, and t ∈ I with [t − 2v_{0}, t + 2v_{0}] ⊆ I. Then, both p̃λ̃|[t − 2v_{0}, 0] and p̃λ̃|[0, t + 2v_{0}] must intersect p̃λ̃′|[t − 2v_{0}, t + 2v_{0}]. Since trainpaths in τ̃ have connected intersection, it follows that p̃λ̃(t) ∈ p̃λ̃′|[t − 2v_{0}, t + 2v_{0}]. Moreover, p̃λ̃(I) and p̃λ̃′(I) intersect positively there. Less formally this is saying that, up to changing the parametrisations by a bounded amount, p̃λ̃ and p̃λ̃′ agree in all but a bounded neighbourhood of the boundary of I.

Reinterpreting in terms arcs in γ, we have shown:

Corollary 3.4 : There is some T_{0} such that, with same hypotheses as Theorem 3.3, if x ∈ ζ is at least a distance T_{0} from the endpoints of ζ, then there exist y, z ∈ ζ′ with d_{γ}(y, z) ≤ T_{0}, d(q̃x̆, q̃y̆) ≤ h_{0}, p̃(x̆) = p̃(z̆), and with ζ̆ and ζ̆′ intersecting positively at p̃(x̆) = p̃(z̆). ♢

4. Quasiprojections.

We want to associate to a curve, α ∈ X(Σ), another “derived curve”, α′ ∈ X(Σ), with l_{M}(α′) bounded. This will lead to a quasiprojection on the curve graph G(Σ), giving us a proof of Theorem 0.2. The derived curve is obtained surgery on the original. We begin with the relevant topological construction.

By a bridge arc, we mean an arc, ϵ, embedded in Σ with endpoints, x, x′ ∈ α. We regard it as defined up to homotopy relative to its endpoints. We realise it so that |α∩ϵ| is minimal. We can think of α ∪ ϵ as the immersion of a theta-curve in Σ. If θ and θ′ are the components of α \ {x, x′}, we refer to the closed curves ϵ ∪ θ and ϵ ∪ θ′ as the halves of the theta-curve. Now, if we put an orientation on α, then there is a parity to each intersection of ϵ with α. If the parities at the endpoints, x, x′, are the same, then we say that ϵ is consistently oriented.

Definition : We say that ϵ, θ is simple if it is consistently oriented and ϵ ∩ θ = {x, x′}.

In this case, ϵ ∪ θ is an essential non-peripheral simple closed curve — the derivative of ϵ.

Definition : We say that ϵ, θ is semisimple if it is consistently oriented, and ϵ ∩ θ = {x, x′, y} with y ∉ {x, x′}, and where the intersection at y has opposite partity to that at x (and therefore at x′).

In this case, we need a more complicated surgery. Let ϵ = ϵ_{1}∪ϵ_{2} and θ = θ_{1}∪θ_{2}, where ϵ_{1}, θ_{1} are the segments between x and y and ϵ_{2}, θ_{2} are the segments between y and x′. We now set the derivative to be θ_{1} ∪ ϵ_{2} ∪ (−θ_{2}) ∪ (−ϵ_{1}). In other words, we are performing a surgery at y to eliminate the intersection there. Note that the derivative is again simple, essential and non-peripheral.

We note:

Lemma 4.1 : Suppose that ϵ is a consistently oriented bridge arc of α, and that θ is a component of α \ ∂ϵ. There is a subarc, ϵ′ ⊆ ϵ, that is also a bridge arc of α, and a component, θ′, of α \ ∂ϵ′, with θ′ ⊆ θ, and such that ϵ, θ is either simple or semisimple.

Proof : This follows easily considering the intersections of θ along ϵ. ♢

Suppose now that q : α ⟶ M is a realisation of α, and that α is not short, so that each component of q̃: α̃ ⟶ M̃ is uniformly quasigeodesic. Suppose that ζ, ζ′ are arcs in α, and that ϵ is a bridge arc connecting x ∈ ζ to x′ ∈ ζ′. We refer to ζ and ζ′ as terminal arcs. We say that ζ and ζ′ are U-long if each component of ζ \ {x} and of ζ′ \ {x′} has d_{γ}-length at least U. If ζ̆ is any lift of ζ to α̃, then we can also lift ϵ and ζ′ to ϵ̆ and ζ̆′ respectively, so that ϵ̆ connects x̆ ∈ ζ̆ to x̆′ ∈ ζ̆′. If we orient ζ and ζ′ so that they cross ϵ in the same sense (regardless of their orientations in α), then we say that qζ and qζ′ are parallel if q̃ζ̆, q̃ζ̆′ are parallel in M̃. Note that if we were to allow ourselves to slide x and x′ along ζ and ζ′, we could always arrange that dM̃(x̆, x̆′) ≤ h_{0}. In this case, we say that (qζ, qx), (qζ′, qx′) are parallel. We can then extend q : α ⟶ M to q : α ∪ ϵ ⟶ M so that ϵ is a path of length at most h_{0} in the appropriate homotopy class.

In general, we say that a bridge, ϵ, with endpoints x, x′ is U-wide of we can find U-long terminal arcs ζ, ζ′ ⊆ α (not necessarily disjoint), with x ∈ ζ, x′ ∈ ζ′ so that (qζ, qx), (qζ′, qx′) are parallel.

We can extend this terminology to an intersection, y, of α with ϵ. In this case, saying that ϵ is U-wide at y is interpreted to mean that both ϵ_{1} and ϵ_{2} are U-wide, where y cuts ϵ into ϵ_{1} and ϵ_{2}. (This means that y lies in a third arc, ζ′ ⊆ α, again oriented, so that ζ′′ crosses ϵ at y in the same direction as ζ and ζ′, and satisfying the same conditions).

In practice, we will be dealing with the following two situations. Suppose that ϵ is a bridge, and that θ is a component of α \ ∂ϵ. Suppose ϵ, θ is simple and U-wide. Then α′ = ϵ ∪ θ lies in X(Σ), and q : α ∪ ϵ ⟶ M restricts to a map q : α′ ⟶ M. Thus l_{M}(α′) ≤ length(q(α′)) ≤ length(θ) + h_{0}. (Recall that the metric d_{α} is defined as the path-metric induced from M.)

Suppose that ϵ, θ is semisimple, with θ crossing ϵ at y, and that y cuts ϵ into ϵ_{1} and ϵ_{2}. Suppose that ϵ is U-wide at y. We can extend q to a map q : α ∪ ϵ_{1} ∪ ϵ_{2} ⟶ M so that q(ϵ_{1}) and q(ϵ_{2}) both have length at most h_{0}. This also gives us a map of the derived curve α′ into M, and we see in this case that l_{M}(α′) ≤ length(θ) + 2h_{0}.

Definition : We say that ϵ is a U-surgerable bridge if it is one of the above two types.

To describe our procedure, we need to indroduce some constants depending only on the parameters of the action, to be specified later. We will fix a constant, ν_{3} ≤ ν_{1}, and set Δ_{3} = Δ_{ν_{3}}. We will also choose two constants, U_{0} and U_{1}, and set U_{2} = 2U_{0} + 2U_{1}.

Suppose now that α ∈ X(Σ), and that q : α ⟶ M is a realisation. We proceed as follows:

(1) If l_{M}(α) ≤ U_{2}, set α′ = α.

(2) If l_{M}(α) > U_{2} and there is some δ ∈ X(Σ) with q(α) ∩ Δ_{3}(δ) ≠ ∅, set α′ = δ.

(3) Suppose that l_{M}(α) > U_{2} and α ∩ Δ_{3} = ∅. Suppose that ϵ is a bridge arc, and θ is a component of α \ ∂ϵ with length(θ) ≤ U_{1}. Suppose that ϵ, θ is U_{0}-surgerable. Then we set α′ to be the derived curve of ϵ, θ, as described above (depending on whether ϵ, θ is simple or semisimple).

The above cases are clearly mutually exclusive, though it is less clear that they are inclusive, and we postpone the issue of existence until later (see Lemma 4.4).

We begin by showing that a choice of derived curve in the above proceedure is well defined up to bounded intersection in Σ. Indeed, any respective derived curves of two disjoint curves have bounded intersection. In what follows we can suppose that we have fixed a realisation in M for each α ∈ X(Σ) (though this choice makes no essential difference).

We begin by setting U_{0} to be the constant T_{0} of Theorem 3.3, given ν = ν_{3} (to be specified later).

Now if α ∩ Δ_{3} = ∅, we can apply Theorem 3.3 (the uniform injectivity theorem). Let p : α ⟶ τ be a carrying map to an associated track, τ. If ζ, ζ′, ϵ are as above, then the trainpaths p̃σ̆ and p̃ζ̆′ intersect positively. Back down in τ, this means that the trainpaths pζ and pζ′ intersect positively, and moreover, we can extend p to a map of α ∪ ϵ homotopic to the original map of α ∪ ϵ into Σ, such that pϵ is a point of this intersection.

We can now apply this in case (3) of the above construction. If ϵ, θ is simple, we slide ϵ as above, and see that the derived curve, α′ is represented (up to free homotopy in Σ) as a closed trainpath in τ. Similarly, if ϵ, θ is semisimple, then we can slide ϵ_{1} and ϵ_{2} separately, and again see that α′ is represented by a closed trainpath. We denote such a closed trainpath by pα′. Since p is lipschitz, we see that such a closed trainpath has bounded d_{τ} -length. It must therefore also have bounded combinatorial length, following the discussion at the end of Section 3. That is, it passes through a bounded number of branches of τ.

We are now ready for:

Proposition 4.2 : Suppose that α, β ∈ X(Σ) are equal or adjacent in G(Σ), and that α′, β′ are respective derived curves (obtained by the above proceedure). Then the intersection number, ι(α′, β′) is bounded above in terms of the parameters of the action.

Proof : Let γ be the multicurve α ∪ β. The realisations q : α ⟶ M and q : β ⟶ M together give us a realisation q : γ ⟶ M. Let p : γ ⟶ τ be a carrying map to an associated multitrack, and let S be the set of special loops.

Consider first α. Note that by Lemma 3.2, if δ ∈ X(Σ) with q(α) ∩ Δ_{3}(δ) ≠ ∅, then δ ∈ S, and pα′ = pδ has combinatorial length 1. Thus in all three cases, pα′ has bounded combinatorial length. The same goes for pβ′. It now follows that ι(α′, β′) is bounded. ♢

In fact, we have also effectively shown:

Lemma 4.3 : If α′ is any derived curve of α, then ι(α, α′) is bounded above in terms of l_{M}(α) and the parameters of the action.

Proof : This follows from the above argument, since the combinatorial length of pα in τ is bounded above in terms l_{M}(α). ♢

Note that all the above works for an arbitrarily chosen constant, U_{1} ≥ 0. We now move on to consider the existence of derived curves. For that, we will need to choose U_{1} sufficiently large in relation to the parameters of the action.

First we make some general observations regarding bridge arcs.

Suppose that α is a curve and ϵ a bridge arc. Suppose that p : α ⟶ τ is a carrying map that extends to a map p : α ∪ ϵ ⟶ τ, homotopic to the standard map of α ∪ ϵ into Σ, and sending ϵ to a point. Suppose that the endpoints x, x′ of ϵ lie in intervals μ, μ′ in α (not necessarily disjoint). We say that μ and μ′ follow each other in τ if pμ and pμ′ are equal as trainpaths in τ. This means that the lifts, μ̆, μ̆′, to τ̃ get mapped to identical paths, p̃μ̆ = p̃μ̆′, in τ̃. Now suppose that α crosses ϵ at a third point, y. Then there must be an arc, μ′′ ⊆ α containing y, so that μ′′ follows both μ and μ′ in τ. (This is with respect to the bridge arcs ϵ_{1} and ϵ_{2} inside ϵ.) This is due the fact that μ′′ gets trapped between μ and μ′. (More formally, we can think of α as a leaf in a foliated neighbourhood of τ in Σ (cf. [PH]).

In summary, using the earlier discussion of simple and semisimple bridges, this entails the following. Suppose we can find a bridge arc, ϵ, whose endpoints lie at the centre point of an arc in α, of τ-length at least some constant, V_{0}, say and that these follow each other in τ and in the same direction (taking induced orientations from an orientation of α). Let θ be a component of α \ ∂ϵ. Then we can assume that ϵ, θ is either simple or semisimple. In the second case, we can find another arc, ζ′′, centred on the third intersection of ϵ with θ, oriented in the opposite direction along α, which also follows ζ and ζ′ in τ, and so also has τ-length at least V_{0}.

Note that we have seen that the map p̃: α̃ ⟶ τ̃ is a quasi-isometry on each component of α̃. Thus, if V_{0} is sufficiently large, we can assume that the d_{α}-lengths of μ, μ′ (and μ′′) are as large as we want. Moreover, since they follow each other in τ, the paths qμ, qμ′ (and qμ′) must be parallel in M over most of their length. More precisely, there are subarcs ζ ⊆ μ, ζ′ ⊆ μ′ (and ζ′′ ⊆ μ′′) so that qζ, qζ′ (and qζ′′) are parallel in M, and where each component of μ \ ζ, μ′ \ ζ′ (and μ′′ \ ζ′′) have bounded d_{α}-length. If we choose V_{0} sufficiently large in relation to U_{0}, we see that, by definition, the bridge arc, ϵ (as well as ϵ_{1} and ϵ_{2}) is U_{0}-wide, and so ϵ is, by definition U_{0}-surgerable.

To find such a bridge arc, we proceed as follows. Suppose q(α) ∩ Δ_{3} = ∅. Suppose ω is an arc in α of length U_{1} to be determined below. As discussed at the end of Section 3, this places a lower bound on the combinatorial length of pω in τ. Now there are only a bounded number of combinatorial possibilities for the image in τ of any subarc of ω of length U_{0}. Thus, provided U_{1} is large enough, we can always find two disjoint subarcs, μ, μ′, of ω of length 2U_{0} which follow each other in τ.

With the above choice of U_{1}, we have:

Lemma 4.4 : Each curve in X(Σ) has a derived curve.

Proof : In other words, at least one of the cases (1), (2) or (3) of the construction must always arise.

Let α ∈ X(Σ). By Lemma 3.2, if q(α) ∩ Δ_{3} = ∅, q(α) meet Δ_{3}(δ) for some δ ∈ S. In particular δ ∈ X(Σ).

Thus, if neither case (1) or (2) arise, l_{M}(α) > U_{2} and q(α) ∩ Δ_{3} = ∅. We now let τ be and associated track, and let ω ⊆ α be any arc of length U_{1}. We can now apply the argument above to find a U_{0}-surgerable bridge arc, and so we are in case (3). ♢

Now for any α ∈ X(Σ), we choose any derived curve α′ ∈ X(Σ) and set proj(α) = α′.

Proposition 4.5 :

(1) There is some l_{0} ≥ 0 such that proj(X(Σ)) ⊆ X(M, l_{0}).

(2) There is some r_{0}, such that if α, β ∈ X(Σ) are adjacent in G(Σ), then d_{G}(proj α, proj β) ≤ r_{0}.

(3) Given any l ≥ l_{0}, then there is some r(l) such that if α ∈ X(M, l), then d_{G}(α, proj α) ≤ r(l).

Proof : We know that for any γ ∈ X(Σ), the stable length, l_{M}^{S}(γ) agrees with the shortest length in M, namely l_{M}(γ), up to an additive constant, so it doesn’t really matter which we deal with.

The fact that the length of a derived curve is bounded is an immediate consequence of the construction, thereby proving (1). Parts (2) and (3) follow respectively by Proposition 4.2 and Lemma 4.3. ♢

Proof of Theorem 0.2 : This now follows from the fact that G(Σ) is hyperbolic, and the existence of a quasiprojection with the properties described by Proposition 4.6, cf. [Bo2]. ♢

Before leaving this section we elaborate on the discussion of bridge arcs for reference in Section 6. Here we shall deal will a multicurve γ, and we allow a bridge arc to connect different components of γ.

Let x, x′ be the endpoints of ϵ. Let ζ, ζ′ be the be terminal arcs in γ, i.e. containing x, x′ in their interiors. Let x̆ ∈ ζ̆, x̆′ ∈ ζ̆′ be the respective lifts to γ̃. Recall that ζ and ζ′ are U-long if if each component of ζ \ {x} and ζ′ \ {x′} has d_{γ}-length at least U. We say that ϵ is U-wide if ζ, ζ′ are U-long, if (qζ, qx), (qζ′, qx′) are parallel in M. In particular, this means that d(q̃x̆, q̃x̆′) ≤ h_{0}. We could weaken this last assumption to placing some other bound on d(q̃x̆, q̃x̆′). Provided U is large enough in relation to this bound, then this is equivalent to saying that we can we can slide the endpoint x′ along ζ to some other point y ∈ ζ so that d(q̃x̆, q̃y̆) ≤ h_{0}. In other words, (qζ, qx), (qζ, qy) are parallel. In so doing, we reduce the width of the bridge arc by at most a bounded amount. This leads to:

Definition : The bridge arc ϵ is h-almost U-wide if we can slide its endpoints a d_{γ}-distance at most h along γ so that the resulting bridge arc is U-wide.

Suppose now that q(γ) ∩ Δ_{ν} = ∅. Let τ be the associated track. There is an upper bound on the total length of (τ, d_{τ}). Moreover, since f(τ) ∩ Δ_{2} = ∅, we see that there is a positive lower bound on the length of any closed curve in τ (not necessarily a trainpath) that is essential in Σ. Now, after splitting τ along a set of short arcs, we can assume that there is a positive lower bound on the length of each branch of τ. These bounds depend ultimately on ν and the parameters of the action. Since we are only interested in d_{τ} up to uniform bilipschitz equivalence, we may as well assume that each edge of τ has unit length. In other words, we are taking the combinatorial distance on d_{τ}.

Suppose that ϵ is bridge arc of γ.

Definition : We say that ϵ is W -wide in τ if there are W -long terminal arcs ζ, ζ′ such that pζ and pζ′ follow each other in τ, and if we can extend p to a map γ ∪ ϵ in the given homotopy class so that ϵ gets mapped to a point. We say that ϵ is h-almost W -wide if we can slide one of its endpoints a d_{γ}-distance at most h in γ such that the resulting bridge arc is W -wide. We say that it is almost W -wide if it is h-almost W -wide for some h.

We note:

Lemma 4.6 : There is some W_{0} depending only on the parameters of the action such that the following holds. Suppose that q : γ ⟶ M is a realisation and that q(γ) ∩ Δ_{3} = ∅. Let τ be an associated track with combinatorial metric, and carrying map p : γ ⟶ τ. Let ϵ be a bridge arc, and let W ≥ 0. If ϵ is (W + W_{0})-wide in τ, then it is W_{0}-almost W -wide in M. If ϵ is (W + W_{0})-wide in M, then it is W_{0}-almost W -wide in τ.

Proof : This follows from the earlier discussion, now using Corollary 3.4 in place of Theorem 3.3. ♢

5. Surfaces of bounded geometry.

In this section we describe some basic facts regarding curves on hyperbolic surfaces of bounded geometry, that is, with injectivity radius bounded below. In Section 6, we will apply this to give a formulation of the Uniform Injectivity Theorem in the case where the realisation of a multicurve in M lies in the thick part.

We begin with a purely topological discussion. We say that a subsurface Φ ⊆ Σ is incompressible if no component of Φ is a disc, no two annular components are freely homotopic, and no component of Σ\Φ is either a disc or a peripheral annulus. (We are not assuming that Φ to be connected.) We write F to the set of incompressible surfaces up to homotopy in Σ. We write ∂Φ for the intrinsic manifold boundary of Φ, and ∂_{Σ}Φ = ∂Σ \ Φ for the relative boundary in Σ.

We write X(Σ, Φ) for the set of curves of X(Σ) that can be homotoped into Φ. We can identify this with X(Φ) ∪ X(∂_{Σ}Φ), where X(∂_{Σ}Φ) is the set of componenents of ∂_{Σ}Φ defined up to homotopy in Σ. We can partially order Φ by inclusion up to homotopy.

The following is a fairly simple observation:

Lemma 5.1 : If Z ⊆ X(Σ) is any set, then there is a unique minimal Φ ∈ F with Z ⊆ X(Σ, Φ). ♢

In this case, we write Φ(Z) = Φ.

Suppose Q, Q′ ⊆ Σ.

Definition : By an ambient map from Q to Q′, we mean a continuous map, ϕ, from Q to Q′, equipped with a continuous extension from Σ to itself, defined up to homotopy relative to Q, and freely homotopic to the identity on Σ.

This is essentially equivalent to saying that ϕ has a preferred lift, ϕ̃ : Q̃ ⟶ Q̃′, between the preimages of Q and Q′ in Σ.

We now fix, once and for all, some η_{0} > 0. By a thick metric on Σ, we mean a hyperbolic metric, σ, on Σ, with geodesic boundary, and with injectivity radius at least η_{0}. This is taken to imply that there is no essential arc of length at most η_{0} between boundary components. All we require of η_{0} is that such a metric should always exist. (It can be defined independently of type(Σ).) Note that the length of the boundary of Σ is necessarily bounded. If x, y ∈ Σ with σ(x, y) ≤ η_{0}, we write [x, y] for the unique shortest geodesic between them.

Suppose η ≤ η_{0}. By an η-sequence in Q, we mean a sequence, x_{0}, x_{1}, …, x_{n}, with x_{i} ∈ Q and σ(x_{i}, x_{i+1}) ≤ η for all i. This determines a broken geodesic η-path, [x_{0}, x_{1}] ∪ [x_{1}, x_{2}] ∪ ⋯ ∪ [x_{n−1}, x_{n}]. We similarly define a cyclic η-sequence and associated closed path. We write Z(Q, η) ⊆ X(Σ) for the set of homotopy classes of all closed simple η-paths with vertices in Q. We set Φ(Q, η) = Φ(Z(Q, η)). In fact, it turns out that X(Σ, Φ(Q, η)) = Z(Q, η).

We can give a more intuitive description of Φ(Q, η) as follows. Take a small regular neigbourhood of N(Q, η/2), and throw away any disc components. (In any case, there will be no disc components when we actually apply this construction.) Now add in all complementary discs and peripheral annuli. Then identify any pair of homotopic annular components. (Again this last step will not be necessary in applications.) The resulting surface agrees with Φ(Q, η) up to homotopy.

Suppose h : [0, ∞) ⟶ [0, ∞) is a homeomorpism, and θ : (Q, σ) ⟶ (Q′, σ′) is a map between metric spaces.

Definition : We say that θ is h-continuous if for all x, y ∈ Q, σ′(θ(x), θ(y)) ≤ h(σ(x, y)). It is h-bicontinuous if it is bijective, and both θ and θ^{−1} are both h-continuous.

Suppose that Q, Q′ ⊆ Σ and σ, σ′ are both thick metics on Σ. Suppose that ϕ : Q ⟶ Q′ is h-continuous. If (x_{i})_{i} is any η-sequence in Q, then (ϕ(x_{i}))_{i} is an h(η) sequence in Q′. If h(η) ≤ η_{0}, then Y (Q, η) ⊆ Y (Q′, h(η)) and so Φ(Q, η) ⊆ Φ(Q′, η′). In particular, if ϕ is h-bicontinuous, we have

Y (Q′, h^{−1}(η)) ⊆ Y (Q, η) ⊆ Y (Q′, h(η))

and

Φ(Q′, h^{−1}(η)) ⊆ Φ(Q, η) ⊆ Φ(Q′, h(η)).

Definition : If I ⊆ R, we say that Q is I-stable if Φ(Q, t) = Φ(Q, u) for all t, u ∈ I.

In this case, we write Φ(Q, I) = Φ(Q, t) for any t ∈ I.

Thus, in the above discussion, if Q′ happens to be [h^{−1}η, hη]-stable with respect to the metric σ, then Φ(Q, η) = Φ(Q′, [h^{−1}η, hη]).

Suppose that γ is a multicurve. We can realise γ as a disjoint union, γ_{σ}, of closed geodesics in Σ, and we write l_{σ}(γ) for its total length.

We will want to compare the realisation of a multicurve, γ, in two different thick structures, σ and σ′, on Σ. We can speak of the “quasi-isometric” distance between σ and σ′ as the minimal quasi-isometric constant of an equivariant quasi-isometry between the covers (Σ̃, σ̃) and (Σ̃, σ̃′). Since we are only interested in bounding this quanitity, we could equivalently speak about bilipschitz or quasi-conformal maps etc., though quasi-isometries are natural to deal with in this context.

Note that if θ : γ_{σ} ⟶ γ_{σ′} is an ambient map, then the lift θ̃: γ̃_{σ} ⟶ γ̃_{σ′} is a quasiisometry with respect to the metrics σ̃ and σ̃′. Thus, we can also view θ̃ as a quasi-isometry between (Σ̃, σ̃) and (Σ̃, σ̃′). Conversely, we can always find such an ambient map, θ, such that the quasi-isometry constants of θ̃ are bounded in terms of the quasi-isometric distance between σ and σ′.

To see this, start with an equivariant quasi-isometry, ϕ : (Σ̃, σ̃) ⟶ (Σ̃, σ̃′) which we can assume continuous. Any point, x ∈ γ̃_{σ} lies in a leaf, μ ⊆ σ̃. Now ϕ(μ) lies a bounded distance from μ′ for some leaf, μ′ of γ̃_{σ′}. Let θ̃(x) be the nearest point to x in μ′. We see that θ̃: γ̃_{σ} ⟶ γ̃_{σ′} is continuous and Γ-equivariant, and that d(x, θ̃(x)) is bounded above in terms of the original quasi-isometry constants. Thus θ̃ is a uniformly quasi-isometry. It descends to the required map θ : γ_{σ} ⟶ γ_{σ′}.

Another observation is that we can assume θ to be uniformly bilipschitz on each leaf. This follows from the fact that any self-quasi-isometry of the real line is a bounded distance from a uniformly bilipschitz map, and moreover, this construction can be made equivariant with respect to infinite cyclic actions. We also need the fact that the ratio of lengths of γ_{σ} and γ_{σ′} are bounded in terms of the quasi-isometric distance. These facts are all well known. In fact, one can do better:

Lemma 5.2 : Suppose that γ_{σ} and γ_{σ′} are realisations of the multicurve γ with respect to two thick structures, σ and σ′ on Σ. Then there is an ambient map θ : γ_{σ} ⟶ γ_{σ′}, which is h-bicontinuous with respect to the metrics σ and σ′, and ξ-bilipschitz with respect to the induced path metrics, where h and ξ depend only on the quasi-isometric distance between σ and σ′.

We begin with a few remarks.

The bilipschitz statement is a local property intrinsic to the curves γ_{σ}, whereas hbicontinuity refers to the ambient metrics, and therefore makes reference to the “transverse structure” of the curves. In fact, we can take h to have the form [t ↦ at^{b}], where a, b > 0 depend only on the quasi-isometric distance. In other words, θ is uniformly biholder.

In particular, the map θ̃ is a uniform quasi-isometry. Note that any two equivariant quasi-isometries remain a bounded distance apart. So the statement can be interpeted as saying that, if we start with an ambient map, then we can homotope it a bounded amount so that it becomes biholder and intrinsically bilipschitz, where the constants depend only on the quasi-isometry constants of the lift of the original map.

Proof of Lemma 5.2 : We split the proof into two stages. We first construct a map that is uniformly biholder, dropping the bilipshitz requirement on γ. For this, we use a fairly standard construction.

If σ and σ′ are two thick structures on Σ then there is a uniformly biholder map between the respective projectives tangent bundles, P Σ and P Σ′. First, lift to H^{2}. Suppose that λ is a tangent line at a point x ∈ H^{2}. Consider the endpoints, y_{1}, y_{2} ∈ ∂H^{2}, of the bi-infinite geodesics tangent to λ at x, and those, y_{3}, y_{4}, of the othogonal bi-infinite geodesics. There is a unique homeomorphism ∂H^{2} ⟶ ∂H^{2}, equivariant with respect to the covering transformations of (Σ, σ) and (Σ, σ′). Let y′_{1}, y′_{2}, y′_{3}, y′_{4} be the respective images of y_{1}, y_{2}, y_{3}, y_{4} under this homeomorphism. The geodesics [y′_{1}, y′_{2}] and [y′_{3}, y′_{4}] intersect at some point, x′ ∈ H^{2}. The angle between them is bounded below by some positive constant depending only on the type of Σ and the quasi-isometric distance between σ and σ′. Let λ′ be the tangent line to [y′_{1}, y′_{2}] at x′. The map between projectivised tangent bundles that sends λ to λ′ is biholder. It descends to a biholder homeomorphism from P Σ to P Σ′. The constants depend only on Σ and the quasi-isometric distance. Note that it is invariant under tangential flow. In particular, it gives a biholder map from γ_{σ} to γ_{σ′}.

We need to modify this so that it is intrinsically bilipschitz on γ_{σ}. To this end, we find constants 0 < η_{2} < η_{1} < η_{0}, η_{3} > 0, h_{0} ≥ 0 and m ∈ N, depending only on the topological type of Σ, such that there is some set, I, of at most m geodesic segments in (Σ, σ), each of length at most η_{1}, with the following properties. If I ∈ I, then it meets γ_{σ} transversely at angles at least η_{3}, and each component of I \ γ containing and endpoint of I has length at least η_{2}. Moreover, any two distinct elements of I are a distance at least η_{2} apart, and each component of γ_{σ} \ ⋃ I has length at most h_{0}. The existence of such an I follows by considering a small metric neighbourhood, N, of γ_{σ} in (Σ, σ). This will be foliated by geodesics segments of this type, transverse to γ_{σ}. Since the topology of N is bounded, we restrict to a bounded number of such segments. This is all fairly standard technology developed in relation to train tracks (cf. [PH]). We now modify the map from γ_{σ} to γ_{σ′} so that it agrees with the original on γ_{σ} ∩ ⋃ I, and interpolate linearly on the components of γ_{σ} \ ⋃ I. It is easily seen to remain biholder and it becomes intrinsically bilipschitz on γ_{σ}. This then proves Lemma 5.2. ♢

Now suppose that we have an intrinsic path-metric, d_{γ}, on γ, and an ambient map j : γ ⟶ γ_{σ} ⊆ (Σ, σ), that is ξ_{0}-bilipschitz for some fixed constant ξ_{0} (in applications depending only on the parameters of the action). On γ_{σ}, we are taking induced pathmetric from σ.

Suppose that ϵ is a bridge arc, with endpoints, x, x′ lying in terminal arcs ζ, ζ′ (as usual, consistently oriented with respect to ϵ, and irrespective of their relative orientation in γ). Let x̆, x̆′, ζ̆, ζ̆′ be their respective lifts to γ̃. Following the terminology of Section 4, we say that jζ and jζ′ are parallel if j̃ζ̆, j̃ζ̆′ are at most a unit distance apart in Σ̃ (in the sense that their respective initial and final points are unit distance apart). Of course, the choice of unity to bound distance is arbitrary here.

Definition : Given V ≥ 0, we say that ϵ is V -wide in (Σ, σ) if we can find V -long terminal arcs, ζ, ζ′ containing the endpoints x, x′ so that jζ and jζ′ are parallel and σ̃(j̃x̆, j̃x̆′) ≤ 1. We say that ϵ is h-almost V -wide if we can slide its endpoints a distance at most h along γ so that it becomes V -wide. We say that it is almost V -wide if it is h-almost V -wide for some h.

Note that being almost V -wide depends only on the homotopy class of ϵ relative to γ. Now suppose that Q ⊆ γ is a closed subset.

Definition : By a bridge path (with respect to Q) we mean a closed path in Σ that is homotopically simple and non-trivial in Σ, and has the form α_{1} ∪ ϵ_{1} ∪ α_{2} ∪ ϵ_{2} ⋯ ∪ α_{n} ∪ ϵ_{n}, where each α is an interval in γ, and each ϵ_{i} is a bridge arc (with endpoints in Q). We say that it is V -wide if each ϵ_{i} is V -wide. We say that it is h-almost V -wide if each ϵ_{i} is h-almost V -wide.

Let Z_{σ}(Q, V, h) be the set of h-almost V -wide bridge paths, and Z_{σ}(Q, V) be the set of all almost V -wide bridge paths. We regard this as defined up to homotopy in Σ, so that Z_{σ}(Q, V, h), Z(Q, V) ⊆ X(Σ). Let Φ_{σ}(Q, V, h) = Φ(Z_{σ}(Q, V, h)) and Φ_{σ}(Q, V) = Φ(Z_{σ}(Q, V)). Recall the notation Φ(Q, η) defined for η > 0, defined earlier in this Section.

Lemma 5.3 : (∀V ≥ 0)(∃η(V) > 0) and (∀η > 0)(∃V (η) ≥ 0) such that if j : γ ⟶ γ_{σ} ⊆ Σ is a uniformly bilipschitz embedding, then

Φ_{σ}(γ, V (η)) ⊆ Φ(γ_{σ}, η)

and

Φ(γ_{σ}, η(V)) ⊆ Φ_{σ}(γ, V).

Here the functions [η ↦ V (η)] and [V ↦ η(V)] depend only on type(Σ) and the bilipschitz constant.

Proof : This follows from the standard fact of hyperbolic geometry that two disjoint geodesic segments in the plane remain close over a long distance if and only if they are exponentially close near their midpoints. ♢

There is a similar statement for a subset Q ⊆ γ.

Lemma 5.4 : Suppose that Q ⊆ γ and h ≥ 0, then there is some r(h) ≥ 0 such that if each component of Q has length at least r(h), then for all η > 0, V ≥ 0, we have

Φ_{σ}(Q, V (η), r(h)) ⊆ Φ(j(Q), η)

and

Φ(j(Q), η(V)) ⊆ Φ_{σ}(Q, V, h),

where the functions [η ↦ V (η)] and [V ↦ η(V)] are as in Lemma 5.3. ♢

We want to relate this to multitracks. We shall assume here that each multitrack, τ, in Σ is given the combinatorial metric, so that each branch has unit length. We note:

Lemma 5.5 : Given any multitrack τ in Σ, there is a thick metric, σ, on Σ and a realisation, τ ↪ Σ, such that the lift, τ̃ ↪ Σ̃, is a unformly quasi-isometric embedding.

Here “uniform” means that the constants of quasi-isometry depend only on type(Σ).

Proof : This is certainly true of any given multitrack, and the result follows from the observation that there are only finitely many combinatorial possibilities for a multitrack, modulo the action of the mapping class group of Σ. ♢

We will refer to an embedding of τ in Σ arising in this way as a uniform embedding.

Lemma 5.6 : Suppose that p : γ ⟶ τ is a uniformly locally bilipschitz carrying map, and that τ ↪ Σ is a uniform embedding with respect to a thick metric, σ, on Σ. Then there is a bilipschitz map, j : γ ⟶ γ_{σ} ⊆ Σ, such that for all x ∈ γ̃, the distance σ̃(p̃(x), j̃(x)) is bounded above. Here, the bounds and bilipschitz constant depend only on type(Σ) and the bilipschitz constant of the carrying map.

Proof : This is similar to the earlier construction. If μ is a leaf of γ̃, then p(μ) is uniformly quasigeodesic in (Σ̃, σ̃), and hence a bounded distance from its realisation as a bi-infinite geodesic μ′. We can thus take nearest point projection, and then approximate by a uniformly bilipschitz map. (Note that we need not worry about the transverse structure here.) ♢

We refer to j as an “associated realisation”.

Suppose ϵ is a bridge arc of γ, and W ≥ 0. Recall the notion of ϵ being W -wide, or h-almost W -wide from the end of Section 4.

Lemma 5.7 : There is some V_{0} ≥ 0, depending only on type(Σ) and ξ_{0} with the following property. Suppose that p : γ ⟶ τ is a uniform embedding, and j : γ ⟶ γ_{σ} ⊆ Σ be an associated realisation. Let ϵ be a bridge arc. If ϵ is (V + V_{0})-wide in τ, then it is V_{0}-almost V -wide in Σ. If ϵ is (V + V_{0})-wide in Σ, then it is V_{0}-almost V -wide in τ.

Proof : Let ζ, ζ′ be terminal arcs of ϵ.

If ϵ is (V + V_{0})-wide in τ, then pζ and pζ′ follow each other in τ, over a distance linearly bounded below by V +V_{0}. The lifts p̃ζ̆ and p̃ζ̆′ to τ̃ are identical. Since τ̃ ↪ Σ̃ is a quasi-isometric embedding and since j̃ : γ̃ ⟶ q̃ is a bounded distance from p̃, we see that j̃ζ̆ and j̃ζ̆′ are quasigeodesic and a bounded distance apart. Thus, their images remain a uniformly bounded distance apart over most of their length, say by 1. In particular, modulo sliding one of the endpoints of ϵ a bounded distance we can assume that σ̃(j̃x̆, j̃x̆′) ≤ 1. It follows that, provided we take V_{0} large enough, then ϵ is V_{0}-almost V -wide in Σ.

Conversely, suppose that ϵ is (V + V_{0})-wide in Σ. The same argument shows that p̃ζ̆ and p̃ζ̆′ as well as p̃x̆ and p̃x̆′, remain a bounded distance apart. We claim that p̃ζ̆ and p̃ζ̆′ follow each other over most of their length. It follows, again choosing V_{0} sufficiently large, that ϵ is V_{0}-almost V -wide in τ. ♢

For the last part of the above argument we need the following observation regarding multitracks.

Lemma 5.8 : Given r ≥ 0, there is some l ≥ 0 depending only on r and type(Σ) with the following property. Suppose that τ ↪ Σ is a uniform embedding and τ̃ ⊆ Σ̃ is its lift to Σ̃. Suppose that μ, μ′ are oriented trainpaths of length at least l in τ̃ that remain distance at most r apart in Σ. Then μ and μ′ intersect positively.

Proof : Let e_{i} and e′_{i} be the ith directed branches of μ and μ′ respectively. For each i, there is some k(i) such that the distance in σ between e_{i} and e′_{k(i)} is at most r. Thus, there are only boundedly many possibilities for the pair (e_{i}, e′_{k(i)}) modulo the covering transformations. Thus, if l is sufficienly large, we can find i < j and a covering translation g with e_{j} = ge_{i} and e′_{k(j)} = ge′_{k(i)}. We can also assume that the distance between e_{i} and e_{j} is large in relation to r. Now the segment, ζ ⊆ μ, between e_{i} and e_{j} and the segment, ζ′, between e′_{k(i)} and e′_{k(j)} project to closed trainpaths in τ that are freely homotopic in Σ and hence equal as closed trainpaths. In particular, k(j) − k(i) = j − i, and ζ and ζ′ lie in the same bi-infinite axis in τ̃ and are oriented in the same sense along this axis. Since their length is greater than r, they must intersect positively. ♢

6. Bounded geometry form of the unform injectivity theorem.

In this section, we give an alternative form of the uniform injectivity theorem in the case where the realisation of our multicurve lies in the thick part of M. This statement makes no reference to tracks, and is more in line with the standard version for hyperbolic 3-manifolds.

We return to the coarse hyperbolic manifold M with Γ ≅ π_{1}(Σ) ≅ π_{1}(M). As in Section 4, we write Δ_{ν} for the thin part Δ_{ν} = Δ(r_{0}, ν).

Proposition 6.1 : Suppose that γ is a multicurve and q : γ ⟶ M is a realisation. Let d_{γ} be the induced path-metric on γ. Suppose q(γ) ∩ Δ_{ν} = ∅. Then there is a bounded geometry structure, σ, in Σ and a map j : γ ⟶ γ_{σ} to the realisation of γ in Σ which is ξ_{0}-bilipschitz in the induced path metric and such that the following hold. Let ϵ be a bridge arc of γ. If ϵ is (U + U_{0})-wide in M, then it is U_{0}-almost U-wide in (Σ, σ). If ϵ is (U + U_{0})-wide in (Σ, σ), then it is U_{0}-almost U-wide in M. Here ξ_{0} and U_{0} depend only on ν the parameters M.

Proof : Let τ be an associated multitrack and p : γ ⟶ τ be the carrying map. Since q(γ) ∩ Δ_{ν} = ∅, as discussed at the end of Section 4, we can assume that τ has the combinatorial metric. Let τ ⟶ (Σ, σ) be a uniform embedding as given by Lemma 5.6, and let j : γ ⟶ γ_{σ} ⊆ Σ be an associated realisation as given by Lemma 5.7.

Now let V_{0} be the constant of Lemma 5.8, let W_{0} be the constant of Lemma 4.6, and set U_{0} = V_{0} + W_{0}. Suppose that ϵ is a bridge arc. If it is (U + U_{0})-wide in (Σ, σ), then it is V_{0}-almost (U + W_{0})-wide in τ, and hence U_{0}-almost U-wide in M. Similarly, if it is (U +U_{0})-wide in M, it is W_{0}-almost (U +V_{0})-wide in τ, hence U_{0}-almost U-wide in (Σ, σ).

♢

We refer to j : γ ⟶ γ_{σ} ⊆ Σ as an associated realisation.

Recall from Section 5 that a bridge path has the form β = α_{1}∪ϵ_{1}∪α_{2}∪ϵ_{2}∪⋯∪α_{n}∪ϵ_{n}, where each α_{i} is an arc in γ and each ϵ_{i} is a bridge. We also assume that it is homotopically simple in Σ (though this makes no essential difference to the discussion). We say that β is U-wide in M if each ϵ_{i} is U-wide.

If Q ⊆ Σ, we define Φ_{M}(Q, U, h) and Φ_{M}(Q, U) exactly as with Φ_{σ}(Q, U, h) and Φ_{σ}(Q, U). An immediate consequence of Proposition 6.1 is the following.

Lemma 6.2 : Let U_{0} be the constant given by Proposition 6.1. Suppose that γ is a multicurve, and q : γ ⟶ M is a realisation with q(γ) ∩ Δ_{ν} = ∅, and let q : γ ⟶ (Σ, σ) be an associated realisation. Then for all U ≥ 0, we have

\begin{aligned}Φ_{σ}(γ, U + U_{0}) & ⊆ Φ_{M}(γ, U) \\ Φ_{M}(γ, U + U_{0}) & ⊆ Φ_{σ}(γ, U). \end{aligned}

Putting this together with Lemma 5.4, we get:

Lemma 6.3 : (∀U ≥ 0)(∃η(U) > 0) and (∀η > 0)(∃U(η) ≥ 0) such that if q : γ ⟶ M and j : γ ⟶ (Σ, σ) are as in Lemma 6.2, then we have

\begin{aligned}Φ(γ_{σ}, η(U)) & ⊆ Φ_{M}(γ, U) \\ Φ_{M}(γ, U(η)) & ⊆ Φ(γ_{σ}, η(U)). \end{aligned}

♢

In particular, suppose η ∈ (0, η_{0}) and set I_{η} = [η(U(η)), η] ⊆ R. Suppose that γ_{σ} is I_{η}-stable in (Σ, σ), then Φ_{M}(γ, U(η)) = Φ(γ_{σ}, I_{η}).

If ϵ is a bridge with endpoints x to x′. We denote by jϵ its realisation in (Σ, σ) as a geodesic with the endpoints fixed at jx and jx′. We write l_{σ}(ϵ) for the length of jϵ.

Suppose that q : γ ⟶ M is a realisation with q(γ) ∩ Δ_{ν} = ∅. Suppose that j : γ ⟶ (Σ, σ) and j′ : γ ⟶ (Σ, σ′) are two associated realisations. We want to consider how they are related.

Lemma 6.4 : There are constants η > 0, l ≥ 0, depending only on the parameters of the action, such that if ϵ a bridge arc with l_{σ}(ϵ) ≤ η, then l_{σ′}(ϵ) ≤ l.

Proof : As in the proof of Lemma 5.5, we see that if η is small enough, then η is (2V_{0})- wide in (Σ, σ). By Proposition 6.1, it is V_{0}-almost V_{0}-wide in M, hence (2V_{0})-almost 0-wide in (Σ, σ′). The last statement means that we can slide the endpoint of ϵ a bounded distance (with respect to the metric d_{γ}, hence also with respect to σ′) so that the geodesic realisation of ϵ in (Σ, σ′) has length at most 1. This implies that its original realisation, j′ϵ, is also of bounded length. ♢

We can rephrase this in terms of the universal cover as follows. Let ϕ : j′ ∘j^{−1} : γ_{σ} ⟶ γ_{σ′}. This has a lift ϕ̃: γ̃_{σ} ⟶ γ̃_{σ′}. Lemma 6.4 says that if x, y ∈ γ̃_{σ} with σ̃(x, y) ≤ η, then σ̃′(ϕ̃x, ϕ̃y) ≤ l.

Our aim is to prove the following:

Lemma 6.5 : Suppose q : γ ⟶ M is a realisation of the multicurve and j : γ ⟶ γ_{σ} ⊆ (Σ, σ) and j′ : γ ⟶ γ_{σ′} ⊆ (Σ, σ′) are associated realisations of γ. Then there is a homeomorphism ψ : γ_{σ} ⟶ γ_{σ′} which is h-bicontinuous and ξ_{0}-bilipschitz with respect to the induced path metric, where the function h and the constant ξ_{0} depend only on the parameters of M. ♢

Note that we have not assumed here that ψ is an ambient map. In fact, the construction will give an equivariant h-bicontinuous lift ψ̃ : γ̃_{σ} ⟶ γ̃_{σ′}. though the associated mapping class of Σ need not be trivial.

For the proof, we need the following construction. If β = α_{1} ∪ϵ_{1} ∪α_{2} ∪ϵ_{2} ∪⋯∪α_{n} ∪ϵ_{n}, and j : γ ⟶ (Σ, σ) is realisation of γ, then we obtain a map, j : β ⟶ Σ, by sending each bridge ϵ_{i} to the corresponding geodesic in (Σ, σ). We say that β is a (η, h)-bridge path if the σ-length of each jϵ_{i} is at most η and the σ-length of each jα_{i} is at most h.

Lemma 6.6 : Let j : γ ⟶ γ_{σ} ⊆ (Σ, σ) be a realisation of γ, and η > 0. Then there exist n ∈ N, depending only in type(Σ) and h > 0, depending only on η and type(Σ) such that each component of ∂_{Σ}Φ(γ_{σ}, η) is homotopic in Σ to a (η, h)-bridge path with at most n-bridges.

Proof : This follows from the description of Φ(γ_{σ}, η) in terms of N(γ_{σ}, η/2). ♢

Corollary 6.7 : Suppose that j, j′ are realisations of the same multicurve, γ and η ≤ η_{0}. Then there is some l ≥ 0, depending only on η and type(Σ) such that l_{σ}(∂_{Σ}Φ(γ_{σ}, η)) ≤ l and l_{σ′}(∂_{Σ}Φ(γ_{σ}, η)) ≤ l.

Here Φ(γ_{σ}, η) is constructed with respect to the metric σ and is defined up to homotopy in Σ.

Proof : The first statement follows directly from Lemma 6.6. For the second, we realise each component of ∂_{Σ}Φ(γ_{σ}, η) as a bridge path β using Lemma 6.6. Applying Lemma 6.4 and the fact that realisations are assumed uniformly bilipschitz with respect to the induced path metrics, we see that the length of the realisation of any such β in (Σ, σ′) has bounded length. Thus, l_{σ′}(∂_{Σ}Φ(γ_{σ}, η)) is bounded. ♢

Note that we can also carry out the above construction in a subsurface, Ψ, of Σ. Suppose that γ ⊆ Ψ. Then γ_{σ} ⊆ Ψ_{σ}, where Ψ_{σ} is realised so that it has geodesic boundary. Given η > 0, we can define Φ_{Ψ}(γ_{σ}, η) intrinsically to Ψ_{σ}. Note that Lemma 6.6 and 6.7 still hold, though the constants will now depend also on l_{σ}(∂Ψ).

Lemma 6.8 : Suppose that j, j′ are realisations of γ, satifying the conclusion of Lemma 6.4 (with constants, η and k). Then there is some l ≥ 0, depending only on η, k, and type(Σ), such that there is a subsurface Ψ ⊆ Σ, with γ ⊆ Ψ, satisfying l_{σ}(∂Ψ) ≤ l, l_{σ′}(∂Ψ) ≤ l, Φ_{Ψ}(γ_{σ}, η) = Ψ and Φ_{Ψ}(γ_{σ′}, η) = Ψ. ♢

Proof : We go back and fore between (Σ, σ) and (Σ, σ′). Let Φ_{1} = Φ(γ_{σ}, η), Φ_{2} = Φ_{Φ_{1}}(γ_{σ′}, η), Φ_{3} = Φ_{Φ_{2}}(γ′_{σ}, η), etc. Thus, Φ_{i} is a descending sequence of subsurfaces of Σ. After a bounded number of steps, there must be three consecutive subsurfaces, all equal to a fixed subsurface Ψ. By the above observation, l_{σ}(∂Φ_{i}) and l_{σ′}(∂Φ_{i}) are bounded inductively, and so in particular, l_{σ}(∂Ψ) and l_{σ′}(∂Ψ) are bounded. By construction, Φ_{Ψ}(γ_{σ}, η) = Φ_{Ψ}(γ_{σ′}, η) = Ψ. ♢

Proof of Lemma 6.5 : Let η, h be the constants of Lemma 6.4. We now apply Lemma 6.8 to give us the subsurface Ψ. Let Ω be a component of Ψ, and let α = γ ∩ Ω. Thus, α has a realisations α_{σ} ⊆ Ω_{σ} and α_{σ′} ⊆ Ω_{σ′}. Let α̃_{σ} ⊆ Ω̃_{σ} and α̃_{σ′} ⊆ Ω̃_{σ′} be the lifts to the universal covers. By construction, any two points of α̃_{σ} and α̃_{σ} are connected by an η-path. Moreover, by Lemma 6.4, if x, y ∈ α̃_{σ} with σ̃(x, y) ≤ η, then σ̃′(ϕ̃x, ϕ̃y) ≤ k, and similarly interchanging α̃_{σ} and α̃_{σ′}. Thus, ϕ̃ is a quasi-isometry. Applying Lemma 5.3 (with Ω in place of Σ) we we get another map, ψ : α̃_{σ} ⟶ α̃_{σ′}, which is uniformly bicontinuous, and bilipschitz in the induced path metrics. We now do this for every component, Ω, of Ψ, and piece them together to give ψ : γ̃_{σ} ⟶ γ̃′_{σ}. Note that if Ω and Ω′ are distinct components of Ψ, then there is a positive lower bound σ(γ_{σ} ∩Ω′_{σ}, γ_{σ} ∩Ω_{σ}) and on σ′(γ_{σ′} ∩Ω_{σ′}, γ_{σ′} ∩Ω′_{σ′})

(arising from the upper bounds on l_{σ}(Ψ) and l_{σ′}(Ψ)). Thus ψ has the required properties.

♢

7. Tight geodesics.

In this section we give proofs of Theorems 0.1 and 0.3 for non-exceptional surfaces. More exactly, we describe how the proofs of the corresponding statements in [Bo2] go through in this more general situation. The statements are essentially identical except that in [Bo2] we assumed that M was a hyperbolic 3-manifold. We made much use of the existence of a supply of “pleating surfaces” in M, in terms of which the uniform injectivity theorem was expressed. The main purpose of this is to relate the geometry in some region of M to the geometry in Σ with some hyperbolic metric. Here we have already established the essential requirements in terms of the uniform injectivity theorem that we have formulated. Indeed that represents most of the work in adapting these arguments. What remains is largely an exercise in translating the terminology.

Let M be a coarse hyperbolic manifold (as defined in Section 0). Recall that X(M, l) = {α ∈ X(Σ) | l_{M}(α) ≤ l}. We have so far shown that X(M, l_{0}) is uniformly quasiconvex in G(Σ) by defining a quasiprojection, proj : X(M) ⟶ X(M, l_{0}). We next recall the definition of a tight geodesic in G(Σ). (This is slight generalisation of that in [MaM] as used in [Bo2].)

If γ is a multicurve, we write X(γ) ⊆ X(Σ) for the set of components of γ. If γ and δ are multicurves, we say that they are exactly distance r apart if d(α, β) = r for all α ∈ X(γ) and β ∈ X(δ). A sequence of multicurves (γ_{i})_{i} is an multigeodesic if for all i < j, γ_{i} and γ_{j} are at exactly distance j − i apart. We say that it is tight at index i if each curve that crosses some curve of X(γ_{i}) also crosses some curve of X(γ_{i−1}) ∪ X(γ_{i+1}). We say that the multigeodesic is “tight” if it is tight for all indices.

A geodesic, (α_{i})_{i} in G(Σ) is defined in the usual way as a multigeodesic whose elements, α_{i}, are all curves. A tight geodesic if there exists a tight multigeodesic, (γ_{i})_{i}, such that α_{i} ∈ X(γ_{i}) for all i. (A complication in this terminology is that a tight geodesic need not be tight when considered as a multigeodesic.)

Let M be a space satisfying (M1)–(M3). We shall fix a realisation, j : α ⟶ α_{M} ⊆ M for each α ∈ X(Σ). By combining these, we get a fixed realisation of each multicurve.

The first thing we need to do is establish analogues of the “tube penetration” lemmas in [Bo2] (these being rephrasings of similar results in [Mi]). In particular, we need versions of Lemmas 5.1, 5.2 and 5.3. A “Margulis tube” (denoted T (δ, η) in [Bo2]) is replaced by a region of the form Δ(δ; r_{0}, ν). Here δ ∈ Y_{0}(Σ) is a primative closed curve in Σ. The number ν ∈ N replaces the “Margulis constant”, η, of T (δ, η). (In the case where M is a hyperbolic 3-manifold, η would be roughly exponential in −ν.) It is no longer clear, a-priori, that the curve δ is simple in Σ, though we shall see this to be so in the cases of interest to us. In the statement of the lemma, the various constants arising will depend on the parameters of the action (and not just on type(Σ) as in [Bo2]).

The strongest form of the tube penetration lemma in [Bo2] was Lemma 5.2, so we give an explicit reformulation of this as follows. Let l_{0} be the constant of Theorem 0.2, so that X(M, l_{0}) is uniformly quasiconvex.

Lemma 7.1 : Given h ≥ 0 and ν ∈ N, there is some ν′ ≥ ν depending only on the parameters of the action and ν, h with the following property. Suppose that δ ∈ Y_{0}(Σ). Suppose that γ_{0}, …, γ_{p} is a multigeodesic which is tight at index i, and with d(γ_{0}, X(M, l_{0})) ≤ h and d(γ_{p}, X(M, l_{0})) ≤ h. If q(γ_{0}) ∩ Δ(δ; r_{0}, ν) ⊆ q(δ) and q(γ_{p}) ∩ Δ(δ; r_{0}, ν) ⊆ q(δ), then q(γ_{i}) ∩ Δ(δ; r_{0}, ν′) ⊆ q(δ).

(Here the statement q(γ) ∩ Δ(δ; r_{0}, ν) ⊆ q(γ) is interpreted to imply that q(α) ∩ Δ(δ; r_{0}, ν) = ∅ for all α ∈ X(γ) \ {δ}.)

Proof : First, note that we can assume that ν ≥ ν_{3}, so that Δ(δ; r_{0}, ν) ⊆ Δ_{3}(δ). Here, ν_{3} is the constant used in defining quasiprojection in Section 4. Thus, if α ∈ X(Σ), and q(α) ∩ Δ(δ; r_{0}, ν) = ∅, then δ ∈ X(Σ) and d(δ, proj(α)) is bounded. (This follows directly from the definition of quasiprojection.) Since X(M, l_{0}) is uniformly quasiconvex, ⋃_{i} X(γ_{i}) lies in a uniform neighbourhood of X(M, l_{0}) depending on h. From standard properties of quasiprojection, it then follows that if α ∈ X(Σ), then d(α, proj(α)) is bounded. In summary, we see that if α ∈ X(γ_{i}) for any i, and if q(α) ∩ Δ(δ; r_{0}, ν) ≠ ∅, then δ ∈ X(Σ) and d(α, δ) ≤ R, where R depends only on ν, h and the parameters of M.

We now find ν′ so that N(Δ(δ; r_{0}, ν′), 2K_{1}(2R + 1)) ⊆ Δ(δ; r_{0}, ν), where K_{1} is the constant in property (2) of an associated track. We claim that this has the desired properties.

For notational convenience, we shift indices to give us a multigeodesic γ_{−m}, …, γ_{0}, …, γ_{r}, which is tight at index 0, and where (q(γ_{−m})∪q(γ_{r}))∩Δ(δ; r_{0}, ν) ⊆ q(δ). Suppose, for contradiction, that there is some α_{0} ∈ X(γ_{0})\{δ} and some y_{0} ∈ α_{0}, with q(y_{0}) ∈ Δ(δ; r_{0}, ν′).

Now certainly q(y_{0}) ∈ Δ_{3}(δ). Let τ_{0} be an associated track to α_{0}. We have maps p_{0} : α_{0} ⟶ τ_{0} and f_{0} : τ_{0} ⟶ M. Now, in Section 3, ν_{1} was chosen so that necessarily f_{0}p_{0}(y_{0}) ∈ Δ_{1}(δ) and f_{0}^{−1}(Δ_{1}(δ)) ⊆ A_{1}(δ), where A_{1}(δ) is the special annular neighbourhood of δ ∈ S_{1} ⊆ S_{0}. From the description of τ_{0} ∩A_{1}(δ), (since A_{1}(δ) is a special annulus), it follows that α_{0} crosses δ. We write A_{1}(δ, τ_{0}) = A_{1}(δ).

Since (γ_{i})_{i} is tight at 0, it follows that either γ_{−1} for γ_{1} also crosses δ. Let σ_{−s}, …, γ_{0}, …, γ_{t} be a maximal sequence of consecutive multicurves, all of which cross δ. Thus, s+t ≥ 1. We claim that either s = m or t = r. For if not, δ ∪ γ_{−s−1} and δ ∪ γ_{t+1} are both multicurves, and so d(γ_{−s−1}, γ_{t+1}) ≤ 2, contradicting the assumption that (γ_{i})_{i} is a multigeodesic. We can thus suppose that t = r. In particular, δ does not lie in γ_{r}. Since, by assumption, q(γ_{r}) ∩ Δ(δ; r_{0}, ν) ⊆ q(δ), we see that, in fact, q(γ_{r}) ∩ Δ(δ; r_{0}, ν) = ∅.

Now let τ_{1} be an associated track for γ_{0} ∪ γ_{1}. We have maps p_{1} : γ_{0} ∪ γ_{1} ⟶ τ_{1} and f_{1} : τ_{1} ⟶ M. Again, f_{1}p_{1}(y_{1}) ∈ Δ_{1}(δ) and f^{−1}(Δ_{1}) ⊆ A_{1}(Σ) = A_{1}(Σ, τ_{1}). Since γ_{1} crosses δ, it follows that τ_{1} ∩ A_{1}(δ) ⊆ p_{1}(γ_{1}). In particular, there is some y_{1} ∈ γ_{1} with f_{1}p_{1}(y_{1}) = f_{1}p_{1}(y_{0}). Now d_{M}(q(y_{0}), f_{1}p_{1}(y_{0})) ≤ K_{1} and d_{M}(q(y_{1}), f_{1}p_{1}(y_{1})) ≤ K_{1}. In particular, d_{M}(q(y_{0}), q(y_{1})) ≤ 2K_{1}. Thus q(y_{1}) ∈ N(Δ(δ; r_{0}, ν), 2K_{1}) ⊆ Δ_{3}(δ).

We can now apply the same argument again, with γ_{1}, γ_{2} replacing γ_{0}, γ_{1} respectively. This time, we find y_{2} ∈ γ_{2} with d_{M}(q(y_{1}), q(y_{2})) ≤ 2K_{1}, and so d_{M}(q(y_{0}), q(y_{2})) ≤ 4K_{1}.

We continue this process inductively. We chose ν′ so that N(Δ(δ; r_{0}, ν′), 2K_{1}(2R + 1)) ⊆ Δ(δ; r_{0}, ν) ⊆ Δ_{3}(δ). Thus, provided i ≤ 2R + 1, we can find y_{i} ∈ γ_{i}, with d_{M}(q(y_{0}), q(y_{i})) ≤ 2K_{1}i. In particular, q(γ_{i}) ∩ Δ(δ; r_{0}, ν) = ∅. But we know that q(γ_{r}) ∩ Δ(δ; r_{0}, ν) = ∅, and so it follows that r ≥ 2R + 1. From the definition of R in the earlier discussion, it follows that d(γ_{0}, δ) ≤ R and d(γ_{0}, δ) ≤ R. Thus, d(γ_{0}, γ_{2R+1}) ≤ 2R, contradicting the assumption that (γ_{i})_{i} is geodesic. ♢

We now move on to consider the construction of subsurfaces. The idea is that if we have a multicurve in Σ that is very long in M, then it fills up a certain subsurface of Σ. One can then see that the subsurfaces associated to consecutive elements of a tight multigeodesic have nice intersection properties, allowing us to shortcut the geodesic, thereby giving a contradiction.

First, we consider a sequence of multicurves, (γ^{n})^{n}, in Σ. Suppose that they have realisations j^{n} : γ^{n} ⟶ γ_{σ}^{n} in (Σ, σ) with respect to a fixed hyperbolic metric, σ. Passing to a subsequence, we can write α^{n} = α̌^{n} ⊔α̂^{n}, where each component of α̌_{σ}^{n} has bounded length, and the length of each component of α̂_{σ}^{n} tends to ∞. Again passing to a subsequence, we can suppose that α̌^{n} = β is fixed, and that α̂^{n} converges on a lamination, λ ⊆ Σ. Let μ(λ) be the union of all minimal sublaminations of λ. We construct subsurfaces F (\underline{α}), G(\underline{α}), H(\underline{α}) as follows. The surface H(\underline{α}) is just a regular neigbourhood of β, G(\underline{α}) is a small neighbourhood of λ with complementary discs and peripheral annuli added in, and F (\underline{α}) is the same thing using μ(λ). Note that F (\underline{α}) ⊆ G(\underline{α}) and G(\underline{α}) ∩ H(\underline{α}) = ∅. This construction is described in Section 3 of [Bo2].

Now, suppose that for each n we have a sequence (α_{i}^{n})_{i} of multicurves, and that α_{i}^{n} is compatible with α_{i+1}^{n} for all i and n (i.e. α_{i}^{n} ∪ α_{i+1}^{n} is a multicurve). We get sequence, \underline{α}_{i} of sequences indexed by n, enabling us to construct a sequence of subsurfaces F_{i} = F (\underline{α}), G_{i} = G(\underline{α}) and H_{i} = H(\underline{α}). These have the intersection properties (F1)–(F9) as laid out out in Section 3 of [Bo2].

Suppose now that M is a coarse hyperbolic manifold. Let Δ_{ν} be the thin part. Suppose that γ is a multicurve and that q(γ) ∩ Δ_{ν} = ∅. There is an associated realisation j : γ ⟶ (Σ, σ) where σ is a thick hyperbolic metric on Σ. Since the properties of an associated realisation are invariant under uniform bilipschitz homeomorphism of Σ, we can always take σ to be fixed modulo the action of the mapping class group of Σ. Put another way, we can fix once and for all the structure, S = (Σ, σ). Then given γ, we can find a geodesic multicurve, α ⊆ S, a mapping class ω sending α to γ, and a new realisation, j′ : γ ⟶ α sending γ to α = (ω^{−1}γ)_{σ}.

In order to maintain consistent notation with [Bo2], we introduce the following maps. We choose a homotopy class χ : M ⟶ Σ which induces the identity on π_{1}(M) ≡ π_{1}(Σ) ≡ Γ. If γ, ω are as above, we write ϕ = χ^{−1} ∘ ω for the corresponding homotopy class S ⟶ M. (Here χ^{−1} is any right inverse homotopy class to χ.) Moreover, we can define ϕ|α to be q ∘ (j′)^{−1} : α ⟶ M. From the property of associated realisations, this is uniformly bilipschitz with respect to the induced path metrics.

This now provides the essential features of a pleating surface, as used in Section 7 of [Bo2] (see the “modified definition” there). In summary, suppose that γ is a multicurve, and that γ_{M} is its realisation in M, and suppose that γ_{M} ∩ Δ_{ν} = ∅ for some ν ∈ N. Then there is a geodesic multicurve, α ⊆ S, and a map ϕ : α ⟶ M, which is uniformly bilipschitz onto γ_{M} in the induced path metric. This extends to a map ϕ : S ⟶ M, defined up to homotopy, so that ω = χ ∘ ϕ : S ⟶ Σ is a homotopy equivalence. The map, ϕ, satisfies the uniform injectivity theorem as laid out in Section 6, where j = (ϕ|α)^{−1} ∘ q is the associated realisation sending γ to α, and q : γ ⟶ γ_{M} is the realisation of γ in M.

The construction of Section 7 of [Bo2] now goes through with little change, as we now describe.

Suppose we have a sequence, M^{n} of spaces with thin parts Δ_{ν}^{n} and maps χ^{n} : M^{n} ⟶ Σ, all defined with respect to the same set of parameters and a fixed value of ν. Suppose that γ^{n} is a sequence of realisations, γ_{M^{n}}^{n} in M^{n} and with γ_{M_{n}}^{n} ∩ Δ_{3}^{n} = ∅. We let α^{n}, ϕ^{n} : Σ ⟶ S be the pleating surface maps described above, and set ω^{n} = χ^{n} ∘ ϕ^{n}. Passing to a subsequence we obtain subsurfaces, F (\underline{α}), G(\underline{α}), H(\underline{α}) in S, and set F^{n} = ω^{n}F (\underline{α}), G^{n} = ω^{n}G(\underline{α}) H^{n} = ω^{n}H(\underline{α}) in Σ.

We want to recognise F^{n}, G^{n}, H^{n} directly in terms of the realisations γ_{M^{n}}^{n} in M^{n}, and without reference to the choice of pleating surfaces. (This is where we made essential use of the uniform injectivity theorem in [Bo2].) To this end, we employ Lemma 6.3. Given η > 0, let I_{η} = [η(U(η)), η]. Now by the process described in [Bo2], we can find some η > 0 (depending on \underline{α}) so that for all sufficiently large n, the realisations α^{n} are I_{η}-stable in S (in the sense described in Section 6 of [Bo2]). Moreover, Φ(α^{n}, I_{η}) = G(\underline{α}) (defined up homotopy). Setting U = U(η), Lemma 6.3 tells us that Φ_{M}(γ, U) = ω^{n}Φ(α^{n}, I_{η}). (Since ϕ^{n}Φ(α^{n}, I_{η}) = Φ(γ_{ϕ^{n}σ}, I_{η}), where γ_{ϕ^{n}σ} denotes the geodesic realisation of γ in the induced structure, ϕ^{n}σ, on Σ. That is we have just introduced a non-trivial mapping class on both sides.) In other words, G^{n} = Φ_{M}(γ^{n}, U). (In [Bo2], this was described in terms of the projectivised tangent bundle. Instead of tangent vectors being close, we now have long segments in γ_{M^{n}}^{n} running parallel. This is merely a reinterpretation of the projectivised tangent bundle to a more general setting.)

To define F^{n}, we proceed similarly. This time, we choose subsets a^{n} ⊆ α^{n}, so that a^{n} ⟶ μ(λ), and let c^{n} = ϕ^{n} ⊆ γ_{M^{n}}^{n}, We again get some η, which we can assume to be the same as the original so that c^{n} is also I_{η}-stable. Moreover, Φ(a^{n}, I_{η}) = F (\underline{α}) and with U = U(η), we get F^{n} = Φ_{M}(c^{n}, U), similarly as above.

We still need to check that this is independent of the choice of c^{n}. But this follows as in Lemma 7.5 of [Bo2], with Proposition 6.5 providing us with the appropriate uniformly bicontinuous homomorphism between different realisations.

We can now apply this construction to a sequence of multicurves. As a result, we can now formulate the analogue of Lemma 7.6 of [Bo2]. Here “(F1)–(F9)” refer to the properties of a sequence of subsurfaces laid out in Section 3 of that paper.

Lemma 7.2 : Suppose we have a sequence, M^{n}, of coarse hyperbolic manifolds with thin parts Δ_{ν}^{n} ⊆ M^{n}, and with χ^{n} : M^{n} ⟶ Σ, all defined with respect to fixed parameters and fixed ν. Suppose, for each n, we have a sequence (γ_{i}^{n})_{i=0}^{n} of geodesic multicurves in M^{n} with γ_{i}^{n} and γ_{i+1}^{n} compatible for all i and n. Suppose that for all i and n, the realisations (γ_{i}^{n})_{M^{n}} satisfy (γ_{i}^{n})_{M^{n}} = ∅. Then, for an infinite subsequence of n, we can construct subsurfaces F_{i}^{n}, G_{i}^{n}, H_{i}^{n} of Σ satisfying (F1)–(F9). ♢

We now move on to Section 8 of [Bo2]. The key lemma is Lemma 8.1 of that paper, which can be rephrased as follows. Let M be a coarse hyperbolic manifold.

Lemma 7.3 : Given k ≥ 0 and ν, p ∈ N, there is some l ≥ 0, depending only on k, p, ν, and the parameters of M such that if (γ_{i})_{i=0}^{p} is a tight multigeodesic in M with (qγ_{0} ∪ qγ_{p}) ∩ Δ_{ν} = ∅, and p ≥ 12r + 19, where r = max{d(γ_{0}, X(M, l_{0})), d(γ_{p}, X(M, l_{0}))}, then there is some i ∈ {0, …, p} such that L(M, γ_{i}) ≤ l.

Proof : We follow the argument in [Bo2]. Assume that the statement fails. We get a sequence of spaces, M^{n}, and sequences (γ_{i}^{n}), of multigeodesics, whose realisations in M^{n} are getting longer and longer. The argument of [Bo2] involves finding suitable Margulis constants with respect to which all the relevant realisations are “non-penetrating”. Here we achieve the analogous statements using Lemma 7.1. We end up with a thin part, Δ_{ν′}, possibly smaller than the original Δ_{ν}. Now Lemma 7.2 (applied with ν′) enables us to construct surfaces (F_{i}^{n}), (G_{i}^{n}), (H_{i}^{n}) and arrive at a contradiction exactly as in [Bo2]. ♢

The same reasoning also applies to the “interpolation lemma”, namely Lemma 8.2 of [Bo2], where “hyperbolic 3-manifold” is replaced by “coarse hyperbolic manifold”.

Theorems 0.1 and 0.3 of this paper are, respectively, reinterpretations of Theorems 1.3 and 1.4 of [Bo2]. What remains of their proofs, as described in Section 8 of [Bo2], now goes through in exactly the same way.

8. Exceptional surfaces.

In this section, we deal with the case where Σ is a one-holed torus or four-holed sphere (abbreviated to 1HT and 4HS respectively). In this case, G(Σ) is interpreted as the modified curve graph with vertex set V (G) = X(Σ), and where α, β ∈ X(Σ) are adjacent if they have minimal possible intersection number (1 or 2 respectively). Thus G(Σ) is a Farey graph. Here every geodesic is deemed tight. Suppose that Σ is a 1HT or 4HS, and that Γ = π_{1}(Σ) acts on a k-hyperbolic space, H. Given γ ∈ X(Σ), we write l(γ) for its stable length (denoted l_{M}^{S}(γ) previously). We suppose that l(δ) ≤ L_{0} for each component δ of ∂Σ. We write X(l) = {γ ∈ X(Σ) | l(γ) ≤ l}.

We claim that Theorems 0.1, 0.2 and 0.3 hold. Indeed, the assumptions (M1) and (M2) are not needed here, and we will restate (M3) explicitly. Thus, for example, Theorem 0.1 is a consequence of the following analogue of Theorem 9.3 of [Bo2].

Theorem 8.1 : Given l ≥ 0, there is some l′ depending only on l, L_{0} and the hyperbolicity constant k with the following property. Let γ_{0}, …, γ_{p} be a geodesic in G(Σ) with l(γ_{0}) ≤ l, l(γ_{p}) ≤ l, then l(γ_{i}) ≤ l′ for all i.

Under the same hypotheses we also have the following, from which Theorem 0.2 follows:

Theorem 8.2 : There is some l_{0}, depending only on k and L_{0} such that for all l ≥ l_{0}, the full subgraph on X(l) is non-empty and connected. Moreover, for all l ≥ l_{0}, X(l) lies in an r-neighbourhood of X(l_{0}), where r depends only k, L_{0} and h.

To relate the above to the case of non-exceptional surfaces, we note that any connected subgraph of a Farey graph is 1-quasiconvex.

There is also a similar variation on Theorem 0.3, which is an immediate consequence of the above, the geometry of the Farey graph, and Lemma 8.3 below.

The logic of the proof will follow that of Section 9 of [Bo2]. For this, we need to reinterpret the essential features of the trace identities used there in terms of inequalities involving distances. We recall from that paper, the following terminology.

Three distinct curves, α, β, γ ∈ V (G) form a triangle if they are pairwise adjacent. Four distinct curves, α, β, γ, δ form a rhombus if α, β, γ and α, β, δ are both triangles.

The key lemma will be:

Lemma 8.3 :

(1) Suppose t ≥ 0 and α, β, γ and form a triangle with l(α) ≤ t and l(β) ≤ t, then l(γ) ≤ t′, where t′ depends only on t, k and L_{0}.

(2) There exists h_{0} ≥ 0 depending only on k such that α, β, γ, δ form a rhombus and l(α) ≥ h_{0}, l(β) ≥ h_{0}, then max{l(γ), l(δ)} ≥ l(α) + l(β) − 2L_{0} − h_{0}.

We have stated this in unified form, whether Σ is a 1HT or 4HS, though the proofs in the two cases are different. (In fact one can give slightly stronger statements in each of the two cases — for example in (2) for a 1HT the term in 2l_{0} can be omitted whereas for a 4HS the assumptions on l(α) and l(β) can be omitted.)

To simpify notation in what follows, we shall introduce the following conventions (cf. [Bo1]). Given x, y ∈ R write x ≃_{t} y, x ⪯_{t} y and x ≪_{t} y to mean respectively |x − y| ≤ t, x ≤ y +t and x+t ≤ y. The constant, t, will change during the course of an argument, but at any given stage will be a fixed multiple of the hyperbolicity constant, h, which could in principle, be determined by following through the argument step by step. For this reason we omit the subscript t. We will behave as though the relations ≃ and ⪯ were transitive, and ⪯ and ≫ were mutually exclusive. This entails choosing a sufficiently large initial constant for ≪ at the outset.

Note that, in these terms, Lemma 8.3(2) could be interpreted as saying that if l(α), l(β) ≫ 0, then max{l(γ), l(δ)} ⪰ l(α) + l(β) − 2l_{0}.

To get from Lemma 8.3 to Theorems 8.1 and 8.2 we proceed as in [Bo2]. Given a rhombus α, β, γ, δ we put a transverse orientation on the edge αβ from γ to δ if l(γ) > l(δ). (If l(γ) = l(δ) we orient it arbitrarily.) Note that if l(α), l(β) ≫ 0, this implies that l(γ) ≫ l(α) + l(β) − 2l_{0}. We note:

Lemma 8.3 : If α, β, γ is a triangle, with αβ and αγ transversely oriented outwards, then min{l(α), l(β), l(γ)} ⪯ 2l_{0}.

Proof : If not, then l(α), l(β), l(γ) ≫ 0, and so l(α) + l(β) ⪯ l(γ) + 2l_{0} and l(α) + l(γ) ⪯ l(β) + 2l_{0}, so l(α) ⪯ 2l_{0} a contradiction. ♢

This is enough to follow through the logic of [Bo2] with l(α) playing the role of traces (after taking logarithms) and inequality interpreted in the obvious way.

To prove Lemma 8.4, we need some general observations about a group Γ acting on a hyperbolic space H.

Given any g ∈ Γ and r ≥ 0, write F (g, r) = {x ∈ H | d(x, gx) ≤ r}. We note that if r ≫ l(g) then F (g, r) is non-empty and uniformly quasiconvex in H. We choose some suitable fixed multiple, h_{1}, of the hyperbolicity constant h, and set F (g) = F (g, l(g) + h_{1}), so that F (g) is always non-empty and uniformly quasiconvex in H. Note that d(x, gx) ⪯ l(g) for all x ∈ F (g). Given g, h ∈ Γ, set D(g, h) = d(F (g), F (h)). If D(g, h) ≫ 0, set σ(g, h) to be a shortest geodesic from F (g) to F (h) (or shortest up to bounded distance). This is uniquely defined up to bounded distance. These statements are all fairly elementary applications of hyperbolicity.

If l(g) > 0, then there is a bi-infinite geodesic (at least up to an arbitarily small additive constant), π(g) with gπ(g) a uniformly bounded distance from π(g). This is welldefined up to uniformly bounded Hausdorff distance. If l(g) ≫ 0 then it is a bounded Hausdorff distance from F (g). We refer to it as an axis of g. If l(g) ≫ 0 this it is assumed to be oriented in the direction of displacement. Note that l(g^{−1}) is the same as l(g) (but oriented in the opposite sense).

If Q ⊆ H and π is a bi-infinite geodesic, we write P_{π}(Q) ⊆ π for the projection of Q to π, i.e. the set of points in π that are nearest to some point in Q. If Q is quasiconvex, then we can assume (up to bounded distance) that P_{π}(Q) is a connected segment of π.

We need one final observation. Suppose that σ is a finite directed segment with length(σ) ≫ 0. Let x, y be it initial and final vertices. We say that g ∈ Γ displaces σ a positive distance t ≥ 0 if there is a geodesic τ from x to gy so that gx and y are a bounded distance from τ and d(x, gx) ≃ t. In this case l(g) ≃ t, and d(y, gy) ≃ t. If t ≫ 0, then the segments σ, gσ and τ all lie in a bounded neighbourhood of π(g). (We can similary define negative displacement, which is equivalent to reversing the orientaion on σ.)

We can use these observations for the following:

Lemma 8.5 : Let g, h ∈ Γ. Then D(g, h) ≫ 0 if and only if l(hg) ≫ l(g) + l(h). In this case, we have:

(1) l(hg) ≃ l(g) + l(h) + 2D(g, h).

(2) σ(g, h) lies in a bounded neighbourhood of π(hg) and is oriented in the same sense.

(3) diam P_{π(hg)}(F (g)) ≃ l(g) and diam P_{π(hg)}(F (h)) ≃ l(h).

Proof : (Sketch) If D(g, h) ≃ 0, then any point x close to both F (g) and F (h) satisfies d(x, gx) ≃ l(g) and d(x, hx) = d(x, h^{−1}x) ≃ l(h). Thus, d(x, hgx) = d(h^{−1}x, gx) ⪯ l(g) + l(h), and so l(hg) ⪯ l(g) + l(h). In other words, l(hg) ≫ l(g) + l(h) implies D(g, h) ≫ 0.

Suppose now that D(g, h) ≫ 0. Let σ = σ(h, g) so that −σ = σ(g, h), and length(σ) = D(g, h) ≫ 0. Consider the three geodesic segments, g^{−1}σ, −σ, hσ, in turn connecting F (g^{−1}hg) to F (g) to F (h) to F (hgh^{−1}). These are all uniformly quasiconvex. From the definition of F (h) we see that σ and hσ must diverge after a bounded distance. It follows that d(F (h), F (hgh^{−1})) ⪰ 2 length(σ) ≫ 0. Similarly, σ and g^{−1}σ diverge after a bounded distance. It now follows that g^{−1}σ, −σ, hσ all lie close to an oriented geodesic. Now hg displaces the long segment g^{−1}σ a long distance to hσ. Thus l(hg) ≫ 0, and g^{−1}σ, −σ, hσ all lie a bounded distance from π(hg). Moreover, d(g^{−1}σ, σ) ≃ l(g) and d(σ, hσ) ≃ l(h). The pojections of F (g) and F (h) to π(hg), up to bounded distance, cover the gaps between the projections of g^{−1}σ, −σ and hσ to π(hg), and diam P_{π(hg)}(F (g)) ≃ l(g) and diam P_{π(hg)}(F (h)) ≃ l(h). We see that the translation distance of hg is approximately length(g^{−1}σ) + l(g) + length(σ) + l(h) ≃ l(g) + l(h) + 2D(g, h). Thus l(hg) ≃ l(g) + l(h) + 2D(g, h). ♢

Suppose that g, h ∈ Γ with l(g) ≫ 0, l(h) ≫ 0 and D(g, h) ≃ 0. We write L(g, h) ∈ R for the distance over which π(g) and π(h) remain almost parallel, taking account of orientations. Thus, L(g, h) = L(h, g) = −L(g^{−1}, h). This well defined up to an additive constant.

Lemma 8.6 : Suppose that g, h ∈ Γ with l(g), l(h) ≫ 0.

(1) If D(g, h) ≫ 0 then l(hg) ≃ l(g) + l(h) + 2D(g, h).

(2) If D(g, h) ≃ 0 then:

(a) if L(g, h) ≥ 0 then l(hg) ≃ l(g) + l(h)

(b) if L(g, h) ≤ 0, then l(hg) ≃ max{|l(g) − l(h)|, l(g) + l(h) + 2L(g, h)}.

In particular, we see that either l(hg) ⪰ l(g)+l(h), or else D(g, h) ≃ 0 and L(g, h) ≪ 0. Part (1) of Lemma 8.6 follows from Lemma 8.5. For part (2) one can first consider the case of a tree. In this case, the equalities

and inequalites hold exactly, as can be seen by consider in turn the various combinatorial possibilites for the axes π(g), π(h) and hπ(g) = π(hgh^{−1}). If σ is a directed segment of π(g) meeting π(h) at its forward endpoints, one considers how this is displaced, in turn, by g and then h. The general case of a hyperbolic space follows by noting that the union of these axes lies a uniformly bounded distance from a tree. In this case, we condider the displacement of a sufficiently long segment σ of π(g) with forward endpoint a bounded distance from π(h) and whose projection to π(h) has bounded diameter. Again, we see that σ is displaced a large positive distance by hg. The displacement agrees with the case of a tree up to an additive constant.

We now split the proof of Lemma 8.3 into two cases.

(A) Σ is a IHT.

Part(1):

Let α, β, γ be a triangle. There are elements a, b ∈ Γ with aba^{−1}b^{−1} peripheral, so that α, β and γ are represented respectively by a, b and ab. We are assuming that l(aba^{−1}b^{−1}) ≤ L_{0}. Suppose that l(a), l(b) ≤ t, and that l(ab) ≫ 2t. By Lemma 8.5 we have D(a, b) ≫ 0. Moreover σ(b, a) lies close to π(ab) and is oriented in the same direction. Also, since D(b, a) = D(a, b) ≫ 0, we have l(ba) ⪰ l(a) + l(b), and σ(a, b) = −σ(b, a) is a bounded distance from π(ba). It follows that π(ab) and π(ba) lie parallel and oriented in opposite directions over a distance at least length(σ(a, b)) = D(a, b). In other words L(ab, ba) ≃ −D(a, b), and so L(ab, (ba)^{−1}) = −L(ab, ba) ⪰ D(a, b) ≫ 0. Thus, by Lemma 8.5, we have l(aba^{−1}b^{−1}) = l(ab(ba)^{−1}) ⪰ l(ab) + l(ba) ≥ l(ab). It follows that l(ab) ⪯ L_{0}.

In summary, we have shown that if l(a), l(b) ≤ t and l(ab) ≫ 2t, then l(ab) ⪯ L_{0}. In other words, if l(a), l(b) ≤ t, then l(ab) ⪯ max{2t, L_{0}}.

Part (2):

Suppose that α, β, γ, δ form a rhombus, They correspond to a, b, ab, ab^{−1}, where a, b ∈ Γ (and aba^{−1}b^{−1} is peripheral). We suppose that l(a), l(b) ≫ 0. We claim that max{l(ab), l(ab^{−1})} ⪰ l(a) + l(b). For if not, l(ab) ≪ l(a) + l(b) and so by Lemma 8.6, D(a, b) ≃ 0 and L(a, b) ≪ 0. Also l(ab^{−1}) ≪ l(a) + l(b) and so L(a, b) ≪ 0. Also l(ab^{−1}) ≪ l(a) + l(b), so L(a, b^{−1}) ≪ 0. But L(a, b^{−1}) = −L(a, b) giving a contradicion.

We have shown that there is a constant k_{0} ≥ 0 such that if l(a), l(b) ≥ h_{0}, then max{l(ab), l(ab^{−1})} ≥ l(a) + l(b) − h_{0}.

(B) Σ is a 4HS.

In this case, if α, β, γ, δ forms a rhombus, then we can find elements p, q, r, s ∈ Γ, each representing a different boundary component of Σ, with pqrs = 1, such that α, β, γ, δ are represented respectively by pq, qr, pr, qs. Note that α, β are also represented by rs and sp respectively. We are assuming that l(p), l(q), l(r), l(s) ≤ L_{0}.

Part (1):

Suppose that α, β, γ forms a triangle, and that l(α), l(β) ≤ t, and that l(γ) ≫ 2t+4L_{0}. Let δ be the curve opposite α across the edge βγ, so that γ, β, α, δ is a rhombus. We can find p, q, r, s ∈ Γ with pqrs = 1 so that γ, β, α, δ are represented respectively by pq, qr, pr, qs. In particular, l(p), l(q), l(r), l(pqr) ≤ L_{0}. Since l(pr), l(qr) ≤ t and l(pq) ≫ 2t + 2L_{0}, it follows that D(p, r) ⪯ t/2 and D(q, r) ⪯ t/2. Let π = π(pq), and set P (q) = P_{π}(F (p)), P (q) = P_{π}(F (q)) and P (r) = P_{π}(F (r)). (Note σ(p, q) is a bounded distance from π(p, q).) Now d(P (p), P (r)) ≃ D(p, r) ⪯ t/2, d(P (q), P (r)) ≃ D(q, r) ⪯ t/2, and d(P (p), P (q)) ≫ t+2L_{0}. It follows that diam P (r) ≫ 2L_{0}. Let τ ⊆ P (r) be a segment with length(τ) ≫ 2L_{0}. This must lie a bounded distance from P (r) and so d(x, rx) ≃ l(r) ≤ L_{0} for all x ∈ τ. Thus there is a subsegment σ ⊆ τ, with length(σ) ≫ 0 such that rσ is a within a bounded neighbourhood of π, oriented in the same sense. Since pq displaces π a distance much more than t + 2L_{0}, we see that pqr displaces σ a distance much more than t + L_{0}. It follows that l(pqr) ≫ t + L_{0} ≥ L_{0}. But l(pqr) ≤ L_{0} giving a contradiction.

In other words, we have shown that if l(α), l(β) ≤ t then l(γ) ⪯ 2t + 4L_{0}.

Part (2):

Let α, β, γ, δ form a rhombus. Let p, q, r, s ∈ Γ be such that pqrs = 1 and l(p), l(q), l(r), l(s) ≤ L_{0} and α, β, γ, δ represented respectively by pq, qr, pr, qs.

We can assume that either l(α) ≫ L_{0} or l(β) ≫ L_{0}, otherwise there is nothing to prove. So we can suppose, without loss of generality, that l(α) = l(pq) ≫ L_{0}.

Suppose, for contradiction, that max{l(pq), l(qs)} ≫ l(pq)+l(qr)−2L_{0}. In particular, we can assume, without loss of generality, that l(pq) ≫ 2L_{0}. Thus, D(p, q) ≫ 2L_{0}. Let π = π(pq) = −π(rs). Set P (p) = P_{π}(F (p)), P (q) = P_{π}(F (q)), P (r) = P_{π}(F (r)) and P (s) = P_{π}(F (s)). By Lemma 8.5, we have diam P (p), diam P (q), diam P (r), diam P (s) ⪯ L_{0}. Also σ(p, q) and σ(s, r) lie a bounded distance from π, are oriented in the same sense, and each has length approximately D(p, q) ≃ D(s, r) ≫ 2L_{0}. We see that the segments of π between P (p) and P (q) and between P (r) and P (s) each have approximately equal length much more than 2L_{0} and oriented in the same sense. It follows that, along π, the most extreme among P (p), P (q), P (r), P (s) are either P (p) and P (r) or else P (q) and P (s).

In the former case, we have D(p, r) ≃ d(F (p), F (r)) ⪰ d(F (p), F (q))+d(F (q), F (r)) ≃ D(p, q) + D(q, r) ≃ l(pq) + l(qr) ≫ 0. Thus l(pr) ⪰ l(pq) + l(pr). In other words, l(γ) ⪰ l(α) + l(β) as required.

Similarly, in the latter case, we have D(q, s) ≃ d(P (q), P (s)) ⪰ d(P (p), P (r)) + d(P (r), P (s)), and so l(qs) ⪰ l(qr) + l(rs) = l(qr) + l(pq). This time, we get l(δ) ⪰ l(α) + l(β). Either way, max{l(γ), l(δ)} ⪰ l(α) + l(β).

Thus, in all cases, we have max{l(γ), l(δ)} ⪰ l(α) + l(β) − 2L_{0}.

This proves Lemma 8.3.

As observed, Theorems 8.1 and 8.2 now follow, applying the logic of [Bo2].

9. Hierarchies.

For applications (in [Bo4]) we need to bound the lengths of all curves in a “hierarchy”. (This is the form of the a-priori bounds theorem for hyperbolic 3-manifolds as given in [Mi].) This is a simple consequence of what we have already done. We state it here explicitly for reference.

Suppose Q ⊆ X(Σ). We construct a larger subset J(Q) ⊆ X(Σ) as follows. If we can find a subsurface Φ ∈ F of Σ with X(∂_{Σ}Φ) ⊆ Q and curves α, β, γ ∈ X(Σ) such that α, β ∈ Q ∩ X(Φ) (identifying X(Φ) ⊆ X(Σ)) and such that γ lies on a tight geodesic from α to β in G(Φ), then we include γ in J(Q). In other words, J_{n}(Q) is the set of all γ arising in this way. Note that Q ⊆ J(Q). (If γ ∈ Q then set Φ = Σ and α = β = γ.) For any n ∈ N, we define J_{n}(Q) inductively by J_{0}(Q) = Q and J_{n+1}(Q) = J(J_{n}(Q)).

(This description differs slightly from that in Section 8 of [Bo2] in that we are allowing Φ to be a 1HT or 4HS in the above, in which case G(Σ) is taken to be the Farey graph.)

For the purposes of stating the result, we can regard a “hierarchy” associated to a subset Q ⊆ X(Σ) to be some canonically defined subset, H(Q), such that H(Q) ⊆ J_{n}(Q) where n depends only on the topological type of Σ.

Theorem 9.1 : Suppose that M is a coarse hyperbolic manifold. Given any k ≥ 0, there is some k′ ≥ 0, depending only k and the parameters of M such that if Q ⊆ X(M, k), then H(Q) ⊆ X(M, k′).

(Recall that X(M, r) = {γ ∈ X(Σ) | l_{M}^{S}(γ) ≤ r}.)

The proof is just an inductive application of Theorem 0.1. Note that if Φ ∈ F with X(∂_{Σ}Φ) ⊆ X(M, r) for some r, then we get a coarse hyperbolic manifold M_{Φ} = M̃/π_{1}(Φ). The parameters of M_{Φ} depend only on those of M. In fact only the parameter associated to (M3) has changed, and that is contolled by the inductive hypotheses.

References.

[Bo1] B.H.Bowditch, Notes on Gromov’s hyperbolicity criterion for path-metric spaces : in “Group theory from a geometrical viewpoint” (ed. E.Ghys, A.Haefliger, A.Verjovsky), World Scientific (1991) 64–167.

[Bo2] B.H.Bowditch, Length bounds on curves associated to tight geodesics : Geom. Funct. Anal. 17 (2007) 1001–1042.

[Bo3] B.H.Bowditch, Train tracks and surface groups acting on hyperbolic spaces : preprint, Warwick (2010).

[Bo4] B.H.Bowditch, Coarse hyperbolic models for 3-manifolds : preprint, Warwick (2010).

[BrCM] J.F.Brock, R.D.Canary, Y.N.Minsky, Classification of Kleinian surface groups II: The ending lamination conjecture : preprint (2004).

[GhH] E.Ghys, P.de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov : Progress in Mathematics No. 83, Birkhäuser (1990).

[Gr] M.Gromov, Hyperbolic groups : in “Essays in group theory”, (ed. S.M.Gersten) M.S.R.I. Publications No. 8, Springer (1988) 75–263.

[H] W.J.Harvey, Boundary structure of the modular group : in “Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference” (ed. I.Kra, B.Maskit), Ann. of Math. Stud. No. 97, Princeton University Press (1981) 245–251.

[MaM] H.A.Masur, Y.N.Minsky, Geometry of the complex of curves II: hierarchical structure : Geom. Funct. Anal. 10 (2000) 902–974.

[Mi] Y.N.Minsky, The classification of Kleinian surface groups I : Models and bounds : preprint, Stony Brook (2002).

[NS] H.Namazi, J.Souto, Revisiting Thurston’s uniform injectivity theorem : in preparation.

[PH] R.C.Penner, J.L.Harer, Combinatorics of train tracks : Annals of Mathematics Studies, No. 125, Princeton University Press (1992).