Hausdorff dimension and dendritic limit sets. Brian H. Bowditch Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, Great Britain. [First draft: January 2003]

Abstract.

Let Γ be a singly degenerate closed surface group acting properly discontinuously on hyperbolic 3-space, H^{3}, such that H^{3}/Γ has positive injectivity radius. It is known that the limit set is a dendrite of Hausdorff dimension 2. We show that the cut-point set of the limit set has Hausdorff dimension strictly less than 2.

0. Introduction.

Let Σ be a closed orientable surface with genus(Σ) ≥ 2, and let Γ = π_{1}(Σ). Suppose that Γ acts properly discontinuously on hyperbolic 3-space H^{3}. Since Γ is torsion-free, the quotient M = H^{3}/Γ is a hyperbolic 3-manifold, with π_{1}(M) ≡ Γ. Its limit set, ΛΓ, is a subcontinuum of the 2-sphere S^{2} ≡ ∂H^{3}. Of particular interest is the case where M is “singly degenerate” so that ∂H^{3} \ ΛΓ is a topological disc. If, in addition, M has positive injectivity radius, inj(M) > 0, then ΛΓ is known to be a dendrite [Min1,Min2]. Moreover, it is shown in [S] that its Hausdorff dimension equals 2.

In contrast, we show:

Theorem 0 : The cut-point set of the dendrite ΛΓ has Hausdorff dimension at most 2 − ϵ, where ϵ > 0 depends only on inj(M) and genus(Σ).

It clearly follows that the set of extreme (i.e. non-cut) points of ΛΓ has Hausdorff dimension 2. (Since every conical limit point is an extreme point, this observation also follows from [BiJ].) In principle, a lower bound on ϵ is computable in terms of inj(M) and genus(Σ), but we shall not make any explicit estimate here. Theorem 0 will follow easily from a stronger statement, namely Theorem 2.1.

This result was inspired by a recent result of Miyachi [Miy] which shows that any pair of points in ΛΓ are connected by a quasi-arc in ΛΓ. A quasi-arc has Hausdorff dimension strictly less than 2, and ΛΓ is a countable union of such arcs. The additional ingredient is thus a uniform bound on these Hausdorff dimensions.

As in [Miy], the essential point is to show that any arc in ΛΓ lies in the boundary of a half-plane quasi-isometrically embedded in H^{3}. Instead of using the full force of the result of [Min2], we proceed directly from the singly degenerate assumption, using ideas of Mitra [Mit], as laid out explicitly in [Bow2].

Dendritic limit sets

1. Quasidendrites.

First we recall some basic facts about dendrites, and introduce the notion of a “quasidendrite” — by analogy with the standard notions of “quasicircle” and “quasi-arc”. (Note that the prefix “quasi-” is used in two different senses in this paper: here from the association with quasiconformal maps, and elsewhere from the association with quasi-isometries.)

There are numerous equivalent definitions of a dendrite. For example, a dendrite, D, is a locally connected metrisable continuum in which any pair of points, x, y ∈ D, are connected by a unique arc, I[x, y] ⊆ D. (Here an “arc” is a subset homeomorphic to a closed real interval.) We write I(x, y) = I[x, y] \ {x, y}. One shows that a point z ∈ D is a cut point if and only there exist x, y ∈ D such that z ∈ I(x, y). We write cut(D) ⊆ D for the set of cut points. A point of D \ cut(D) is called an extreme point.

Since D is metrisable, it contains a countable dense subset P ⊆ D. One can easily show that cut(D) ⊆ ⋃_{x,y∈P} I[x, y]. In particular, cut(D) is a countable union of open subarcs.

Given four points, x, y, z, w in the Riemann sphere, C∪{∞}, we define their crossratio, [x, y : z, w], as \frac{(w−z)(y−x)}{(w−y)(z−x)}. Thus, [x, y : z, ∞] = \frac{y−x}{z−x}.

First recall the notion of a K-quasicircle (cf. [LV]), for K ≥ 1. This can be defined as a simple closed curve, α ⊆ C∪{∞}, such that whenever {x, z} ⊆ α is linked with {y, w} ⊆ α, we have |[x, y : z, w]| ≤ K. Mapping w to ∞ by a Möbius transformation, this means that if β ⊆ α \ {∞} is any compact subarc with endpoints ∂β, then diam(β) ≤ K diam(∂β), where diam denotes euclidean diameter. From this, one can deduce that the Hausdorff dimension of α \ {∞} is at most 2 − ϵ, where ϵ > 0 depends only on K. Since Möbius transformations are smooth, the same applies to α with respect to either the spherical or euclidean metrics.

We can similarly define a quasi-arc, γ, where x, y, z, w are assumed to occur in this order along γ. The same discussion applies. More generally:

Definition : Let D ⊆ C ∪ {∞} be a dendrite embedded in the Riemann sphere. We say that D is a K-quasidendrite if, given any distinct x, y, z, w ∈ D such that y separates x from z and z separates y from w, then |[x, y : z, w]| ≤ K.

Note that the condition on x, y, z, w is equivalent to asserting that y, z ∈ I(x, w) and x, y, z, w occur in this order along I[x, w]. In other words, we are saying that each subarc of D is a K-quasi-arc. Since the cut-point set, cut(D), lies on a countable union of such quasi-arcs, we conclude:

Lemma 1.1 : Given K ≥ 1, there is some ϵ > 0 such that if D ⊆ C ∪ {∞} is a K-quasidendrite, then the Hausdorff dimension of cut(D) is at most 2 − ϵ.

We note that to verify that a dendrite D is a quasidendrite, it is enough, by continuity, to consider x, y, z, w ∈ cut(D).

We also note the following geometric interpretation of the crossratio bound, which is what we will verify in practice. First, identify C ∪ {∞} with boundary, ∂H^{3}, of hyperbolic 3-space, H^{3}. Given x, y ∈ H^{3} ∪ ∂H^{3}, we denote by [x, y] the hyperbolic geodesic from x to y. We write d for the hyperbolic metric on H^{3}. If x, y, z, w ∈ ∂H^{3}, then d([x, z], [y, w]) = | log |μ||, where μ ∈ C \ {0} satisfies 4[x, y : z, w] = μ + μ^{−1} − 2. In particular, we see that an upper bound on |[x, y : z, w]| is equivalent to an upper bound on d([x, z], [y, w]).

2. Kleinian groups.

We recall some basic facts about kleinian groups and give a more precise formulation of our main theorem.

By a kleinian group, we shall mean a group Γ acting freely and properly discontinuously on hyperbolic 3-space, H^{3}. We write ΛΓ ⊆ ∂H^{3} for its limit set, and M = M(Γ) = H^{3}/Γ for the quotient 3-manifold. The injectivity radius, inj(M), of M is twice the infimum of lengths of essential closed curves in M.

An important class of groups arises when Γ = π_{1}(Σ) is the fundamental group of a closed orientable surface, Σ, with genus(Σ) ≥ 2. (The non-orientable case is essentially the same, or can be dealt with passing to a double cover.) In this case, Bonahon [Bon] shows that M is geometrically tame (in particular, homeomorphic to Σ × R). If there are no parabolics, then each end of M is either geometrically finite or simply degenerate. Depending on whether M has 2, 0 or 1 simply degenerate ends, the limit set, ΛΓ, will be respectively, all of ∂H^{3}, a quasicircle, or a continuum, D, with ∂H^{3} \ D a disc. It is the last, “singly degenerate”, case that interests us here (Γ is sometimes called a “Bers group”). It is conjectured that D is always a dendrite, and this is known to be the case if inj(M) > 0 [Min2]. Moreover, under the same hypotheses, Sullivan [S] showed that the Hausdorff dimension of D equals 2. (For a much more general statement, see [BiJ].) In contrast, Miyachi [Miy] showed that every subarc of D is a quasi-arc. However, his argument does not give a uniform constant. We use a slightly different approach to give our main result:

Theorem 2.1 : Let Σ be a closed orientable surface with genus(Σ) ≥ 2. Let Γ = π_{1}(Σ) act on H^{3} as a singly degenerate kleinian group with inj(H^{3}/Γ) > 0. Then the limit set, ΛΓ, is a K-quasidendrite, where K depends only on inj(H^{3}/Γ) and genus(Σ).

Again, the dependence of K on inj(M) and genus(Σ) is, in principle, computable. Putting Theorem 2.1 together with Lemma 1.1, we immediately deduce Theorem 0.

3. Riemannian half-planes.

As in [Miy] we shall prove the main result by constructing quasi-isometric maps of half-planes into H^{3}. We begin with a general discussion of half-planes.

Let H be a complete path-metric space homeomorphic to R×[0, ∞). (We can assume H to be riemannian if we want.) We suppose H to be hyperbolic in the sense of Gromov [Gr,GhH]. We write ∂H for its Gromov boundary, so that H ∪ ∂H is compact. To avoid any confusion, we shall refer to R × {0} as the frontier of H, and denote it by fr(H). We shall assume that the closure of fr(H) in H ∪ ∂H is homeomorphic to a closed real interval with endpoints end(H) = {a, b} ⊆ ∂H. Since H is one-ended, we know that ∂H is connected (see, for example, [GhH]). In fact:

Lemma 3.1 : ∂H is homeomorphic to a closed interval with endpoints end(H).

Proof : Since ∂H is metrisable, it’s enough to show that each point, c, of ∂H \ end(H) separates a from b. To this end, let γ be a geodesic ray with basepoint in fr(H) and converging on c. By cutting γ off at its last intersection point with fr(H), we can assume that γ meets fr(H) in a single point. This point cuts fr(H) into two rays, α and β, converging on a and b respectively.

Now γ cuts H into two half-planes, H_{a} and H_{b}, so that fr(H_{a}) = α ∪ γ and fr(H_{b}) = β ∪ γ. Now H_{a} and H_{b} are convex and hence intrinsically Gromov hyperbolic. We have ∂H_{a} ∩ ∂H_{b} = {c}, ∂H_{a} ∪ ∂H_{b} = ∂H, and a ∈ ∂H_{a} and b ∈ ∂H_{b}. Moreover, ∂H_{a} and ∂H_{b} are both connected. It follows that c separates a from b in ∂H as required.

Lemma 3.2 : Suppose p, q, r, s ∈ ∂H are distinct and that p, q, r, s occur in this order along the interval ∂H. Suppose that α, β ⊆ H are bi-infinite geodesics connecting p to r and q to s respectively. Then α ∩ β ≠ ∅.

Proof : Simple planar topology shows that α bounds a half-plane, H_{α} ⊆ H. This is convex and hence intrinsically Gromov hyperbolic. We have ∂H_{α} ⊆ ∂H and end(H_{α}) = {p, r}. In other words, ∂H_{α} is the subinterval of ∂H with endpoints p and r. Similarly, β bounds a convex half-space, H_{β}, with ∂H_{β} ⊆ ∂H and end(H_{β}) = {q, s}. Note that neither ∂H_{α} nor ∂H_{β} is contained in the other.

Suppose, for contradiction, that α ∩ β = ∅. Now neither H_{α} nor H_{β} is contained in the other, and so H_{α} ∩ H_{β} = ∅. But since end(H_{α}) ∩ end(H_{β}) = ∅, it follows easily that ∂H_{α} ∩ ∂H_{β} = ∅, giving a contradiction. ♢

Before discussing how we apply this, we give some general definitions that will be used again later.

Suppose (X, ρ) and (Y, d) are metric spaces, and let ψ : X ⟶ Y be any map (not necessarily continuous). Let F : [0, ∞) ⟶ [0, ∞) be an increasing function.

Definition : We say that ψ is F -proper if for all x, y ∈ X, ρ(x, y) ≤ F (d(ψ(x), ψ(y))). We say that ψ is coarsely proper if it is F -proper for some function F. (This is sometimes termed “uniformly proper” in the literature.) We say that ψ is quasi-isometric if it is F -proper of some linear function F.

The function F is referred to as the parameter of properness or quasi-isometry.

Given any subset A ⊆ H^{3}, we define its limit set as the intersection of ∂H^{3} with the closure of A in H^{3} ∪ ∂H^{3}.

Lemma 3.3 : Suppose that H is a complete path-metric space homeomorphic to a half-plane. Suppose that ψ : H ⟶ H^{2} is a quasi-isometric map. The limit set of ψ(H) is a K-quasi-arc, where K depends only on the parameters of ψ.

For our applications, we can assume that ψ is continuous, and that ψ restricted to fr(H) extends to a continuous map of the closed interval. This will eliminate a few technical details.

Proof : By the quasi-isometric invariance of hyperbolicity, we see that H is intrinsically hyperbolic, and ψ extends to a continuous map of H ∪ ∂H to H^{3} ∪ ∂H^{3}. The restriction of ψ to ∂H is a homeomorphism from ∂H to the limit set, γ, of ψ(H). Thus, by Lemma 3.1, γ is a closed interval.

Now consider points, x, y, z, w ∈ γ, occurring in this order. These are, respectively, the ψ-images of points p, q, r, s ∈ ∂H. Let α and β be bi-infinite geodesics in H connecting p to r and q to s. By Lemma 3.2, α ∩ β ≠ ∅. But ψ(α) is quasigeodesic in H^{3}, and hence remains a bounded distance from the hyperbolic geodesic [x, z]. Similarly, ψ(β) remains a bounded distance from [y, w]. It follows that d([x, z], [y, w]) is bounded in terms of the parameters of ψ. Thus γ is a quasi-arc as claimed. ♢

4. Simply degenerate ends.

Let Γ be a singly degenerate surface group with quotient M = H^{3}/Γ. Write Y ⊆ H^{3} for the hyperbolic convex hull of ΛΓ, and write core(M) = Y/Γ ⊆ M for the convex core of M. We suppose that inj(M) > 0. (This is sometimes termed “bounded geometry”.)

In this case, Minsky [Min1,Min2], shows that Y is equivariantly quasi-isometric to a certain “singular Sol” model space having a natural singular foliation. (The dependence of the parameters on inj(M) and genus(Σ) is not addressed there, but could, in principle be extracted from the approach discussed in [Bow1,Bow2].) Miyachi [Miy] shows that leaves in this foliation give rise to quasi-isometrically embedded planes. However, his argument does not give uniformity of quasi-arcs. Here we describe another approach, which does give uniformity, and also bypasses much of the difficult part of [Min1,Min2] (namely relating the geometry of core(M) to Teichmüller geodesics, and hence to singular Sol geometry).

Throughout the rest of this paper, we describe a constant or function as “uniform” if it ultimately depends only on the initial data, namely genus(Σ) and inj(M). (It is for this reason that we have substituted the term “coarsely proper” for the more usual “uniformly proper”.)

Now by simple degeneracy and interpolation of pleated surfaces [Bon,T], we can find a sequence of hyperbolic (i.e. constant curvature −1) metrics, ρ_{i}, on Σ and uniformly lipschitz homotopy equivalences, f_{i} : Σ_{i} ⟶ core(M), where Σ_{i} = (Σ, ρ_{i}). The diameters of the images f_{i}(Σ_{i}) are necessarily bounded, and we can assume, moreover, that each f_{i}(Σ_{i}) separates f_{i−1}(Σ_{i−1}) from f_{i+1}(Σ_{i+1}) and that for all i > 0, d(f_{i}(Σ_{i}), f_{i+1}(Σ_{i+1})) is bounded above and below by uniform positive constants. We may as well take the lower bound to be 1. We can also take f_{0} to be an isometry from Σ_{0} to the boundary of core(M). One can show that ⋃_{i=0}^{∞} f_{i}(Σ_{i}) is unformly quasidense in core(M), i.e. each point of core(M) lies within a bounded distance of some f_{i}(Σ_{i}).

Lifting to H^{3}, we obtain a sequence of equivariant uniformly lipschitz maps, ϕ_{i} : X_{i} ⟶ H^{3}, where X_{i} ≅ H^{2} is the universal cover of Σ_{i}. Moreover, it is shown in [Min1] that the maps ϕ_{i} are uniformly coarsely proper. Indeed, the parameters are computable in terms of genus(Σ) and inj(M) (see [Bow1]).

A simple consequence of this, in turn, is that if we choose r > 0 sufficiently large, then the relation, ∼_{r}, defined on X_{i} × X_{i+1} by x ∼_{r} y if d(ϕ_{i}(x), ϕ_{i}(y)) ≤ r is a uniform (depending on r) quasi-isometry from X_{i} to X_{i+1}. By the stability of quasigeodesics, it follows that if x ∼_{r} x′ and y ∼_{r} y′ then ϕ_{i}([x, y]_{i}) is a bounded Hausdorff distance from ϕ_{i+1}([x′, y′]_{i+1}), where [x, y]_{i} denotes the geodesic in X_{i} from x to y. Indeed, there is some uniform s > 0 such that the relation ∼_{s} restricted to [x, y]_{i} × [x′, y′]_{i+1} is a uniform quasi-isometry from [x, y]_{i} to [x′, y′]_{i+1}.

Suppose that x, y, z ∈ X_{i}. We define cent_{i}(x, y, z) ∈ X_{i} to be the nearest point to z in [x, y]_{i}. Another consequence of the above remarks is that if x, y, z ∈ X_{i} and x′, y′, z′ ∈ X_{i+1} with x ∼_{r} x′, y ∼_{r} y′ and z ∼_{r} z′, then d(ϕ_{i}(cent_{i}(x, y, z)), ϕ_{i+1}(cent_{i+1}(x′, y′, z′))) is uniformly bounded in terms of r.

5. Proof of the main theorem.

In this section, we prove Theorem 2.1 by constructing quasi-isometric maps of halfplanes into H^{3}. We begin with a couple of technical observations.

By a riemannian rectangle we shall mean a riemannian metric ρ on the square [0, 1]^{2}, which is isometric to a euclidean metric in a neighbourhood of the “horizontal sides” [0, 1] × {0} and [0, 1] × {1}. We refer to the intervals {t} × [0, 1] as “vertical”.

Suppose that J and J′ are compact real intervals (or path metric spaces homeomorphic, and hence isometric, to compact intervals). Suppose that ∼ ⊆ J × J′ is a quasi-isometry. We note:

Lemma 5.1 : We can identify J and J′ with the horizontal sides of a riemannian rectangle, R, so that each vertical interval has bounded length, so that the inclusions of J and J′ into R are quasi-isometries, and so that the inclusion of J into R composed with the quasi-isometry ∼ agrees up to bounded distance with inclusion of J′ into R.

This can be deduced using the fact that a quasi-isometry of intervals agrees up to bounded distance with a diffeomorphism.

Now suppose that J and J′ are real intervals, and that ϕ : J ⟶ H^{3} and ϕ′ : J′ ⟶ H^{3} are coarsely proper lipschitz maps. Let ∼ ⊆ J × J′ be the relation defined by x ∼ y if d(ϕ(x), ϕ′(y)) ≤ r for some fixed r. Suppose that ∼ is a quasi-isometry. (For example, if HausDist(ϕ(J), ϕ′(J′)) ≤ r where HausDist denotes Hausdorff distance.) Let R be a riemannian rectangle as given by Lemma 5.1. We can extend ϕ⊔ϕ′ to a map ψ : R ⟶ H^{3} by mapping each vertical interval linearly to a geodesic segment in H^{3}. The resulting map ψ will be uniformly coarsely proper. It will also be uniformly coarsely lipschitz, in the sense that d(ϕ(x), ϕ(y)) is bounded above by a uniform linear function of ρ(x, y). (Indeed by constructing the metric on R sensibly, one can assume it to be uniformly lipschitz.)

We shall apply this construction in the situation where we have a sequence of intervals, (J_{i})_{i∈N} and uniformly proper maps ϕ_{i} : J_{i} ⟶ H^{3}, with HausDist(ϕ_{i}(J_{i}), ϕ_{i+1}(J_{i+1})) bounded. The above construction gives us riemannian rectangles, R_{i}, between J_{i} and J_{i+1} and maps ψ_{i} : R_{i} ⟶ H^{3} that are uniformly coarsely lipschitz and coarsely proper. Gluing these together, we get a riemannian half-space, H = ⋃_{i=0}^{∞} R_{i}, and a coarsely lipschitz map ψ : H ⟶ H^{3}. In the case of interest which we describe below, ψ will be quasi-isometric.

We now return to the set-up described in Section 4, where we have a sequence of planes, X_{i} ≅ H^{2}, and uniformly coarsely proper lipschitz maps ϕ_{i} : X_{i} ⟶ H^{3}. It is notationally convenient to assemble the X_{i} into a disjoint union Ξ = ⨆_{i=0}^{∞} X_{i}. We refer to Ξ as a “stack” of “sheets” X_{i}. We write ϕ = ⋃_{i} ϕ_{i} for the map ϕ : Ξ ⟶ H^{3}. Thus ϕ(Ξ) is quasidense in the convex hull, Y, of the limit set D = ΛΓ.

By an r-chain, for r ≥ 0, we mean a sequence \underline{x} = (x_{i})_{i∈N}, with x_{i} ∈ X_{i} and d(ϕ_{i}(x_{i}), ϕ_{i+1}(x_{i+1})) ≤ r for all i. From the separation properties of the images ϕ_{i}(X_{i}), we see that (ϕ_{i}(x_{i}))_{i} is a quasigeodesic sequence in Y, and hence converges on some point, denoted π(\underline{x}), in D.

It is not hard to see that one can find some uniform r_{0} ≥ 0 such that each point of cut(D) has the form π(\underline{x}) for some r_{0}-chain. (In particular, it follows that cut(D) consists entirely on non-conical limit points, as observed in the introduction.) For details, see [Bow2].

Suppose a, b ∈ cut(D) are distinct. We can find r_{0}-chains, \underline{a} and \underline{b} such that a = π(\underline{a}) and b = π(\underline{b}). Let J_{i} = [a_{i}, b_{i}]_{i} and write ϕ_{i} : J_{i} ⟶ H^{3} for the restriction of ϕ_{i} to J_{i}. As observed in Section 4, HausDist(ϕ_{i}(J_{i}), ϕ_{i+1}(J_{i+1})) is uniformly bounded. Thus, we can embed the J_{i} in a riemannian half-plane, H, and construct a map ψ : H ⟶ H^{3} as above. From the construction, ψ(H) ⊆ Y. We denote the metric on H by ρ.

Lemma 5.2 : The map ψ : H ⟶ H^{3} is uniformly quasi-isometric.

Proof : We already know that ψ is uniformly coarsely lipschitz. Thus, given x, y ∈ H, we want to show that d(x, y) is bounded above by a uniform linear function of d(ψ(x), ψ(y)). The argument follows that in [Mit].

Since ⋃_{i} J_{i} is uniformly quasidense in H, we can assume that x, y ∈ ⋃_{i} J_{i}. Let p = ψ(x) = ϕ(x) and q = ψ(y) = ϕ(y). Let α = [p, q] ⊆ Y be the geodesic segment in Y ⊆ H^{3} connecting p to q. Let p_{0} = p, p_{1}, …, p_{n} = q ∈ α cut α into n ≤ d(p, q) + 1 segments, each of length at most 1.

Now ϕ(Ξ) is quasidense in Y, so we can find points x_{i} ∈ Ξ with d(p_{i}, ϕ(x_{i})) uniformly bounded. We can take x_{0} = x and x_{n} = y. We write x_{i} ∈ X_{j(i)}. Since d(ϕ_{j}(X_{j}), ϕ_{j+1}(X_{j+1})) ≥ 1 for all j, we can choose the x_{i} so that |j(i + 1) − j(i)| ≤ 1 for all i.

Now, let z_{i} = cent_{j(i)}(a_{i}, b_{i}, x_{i}) ∈ J_{j(i)}. Note that z_{0} = x_{0} = x and z_{n} = x_{n} = y. Now, d(ϕ(a_{j(i)}), ϕ(a_{j(i+1)})) ≤ r_{0}, d(ϕ(b_{j(i)}), ϕ(b_{j(i+1)})) ≤ r_{0}, and by construction, d(ϕ(x_{i}), ϕ(x_{i+1})) is uniformly bounded. Thus, as observed at the end of Section 5, d(ϕ(z_{i}), ϕ(z_{i+1})) is uniformly bounded. By construction, ϕ(z_{i}) = ψ(z_{i}) and ϕ(z_{i+1}) = ψ(z_{i+1}). Since |j(i + 1) − j(i)| ≤ 1, z_{i} and z_{i+1} lie in the same rectangle, R_{j}, of H. Now ψ restricted to R_{j} is uniformly proper. Thus, ρ(z_{i}, z_{i+1}) is bounded above by a uniform constant, k, say. Thus, ρ(x, y) = ρ(z_{0}, z_{n}) ≤ kn ≤ k(d(p, q) + 1) = k(d(ψ(x), ψ(y)) + 1) as required. ♢

Now, by Lemma 3.3, the limit set, γ, of ψ(H) is a uniform quasi-arc with endpoints a and b. Since ψ(H) ⊆ Y, we see that γ ⊆ D.

We have shown that any pair of points of cut(D) are connected by a uniform quasi-arc in D. It follows that D is a uniform quasidendrite, thus proving Theorem 2.1.

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