We give a geometric construction of the Harer complex for the Teichmuller space of a punctured surface. For a given finite area hyperbolic surface we construct a cut locus. The combinatorics of this cut locus determines the simplex of the Harer complex in which this surface lies as a point in Teichmuller space. The lengths of the projections of the edges of the cut locus onto a fixed set of horocycles determine the coordinates in this simplex. We explain how these coordinates can be extended over all of Teichmuller space, and show that the coordinate transformations are real analytic in a neighbourhood of each simplex, and hence give a real analytic atlas of Teichmuller space.
Topology 27 (1988) 91-117.
A singular euclidean structure on a closed surface is a metric which is locally isometric to the euclidean plane except for a finite number of cone singularities. We descibe a canonical cellutation of the space of singular euclidean structures on a given surface with a given number of cone points. The cells are convex euclidean polyhedra. We also obtain a cellution of the Teichmuller space (times a simplex) of the same surface with the same number of punctures, by exactly the same polyhedral complex. This gives a canonical identification of these spaces, and hence a new proof that the space of singular euclidean structures is topologically a ball.
J. London Math. Soc. 44 (1991) 553-565.
We give criteria for recognising when combinatorial patterns of directed arcs in the disc contain directed cycles. The endpoints of the arcs are imagined as the input and output ports of a planar cicruit, and the arcs represent the allowable logical dependencies of input and output. The motivation is the study of VLSI circuits.
Information and Computation 90 (1991) 178-193.
We give an acount of the notion of hyperbolicity as defined by Gromov in the context of path-metric spaces. We prove the equivalence of five formulations of this notion, including the existence a linear isoperimetric function. We show that any set of $ n $ points in such a space can be approximated up to an additive constant, logrithmic in $ n $, by an immersed tree. We also show that the notion of hyperbolicity propagates, in the sense that a coarsly simply connected path-metric space which is hyperbolic on every ball of a given radius is globally hyperbolic --- provided the various constants are properly quanified.
in ``Group theory from a geometrical viewpoint'' (ed. E. Ghys, A. Haefliger, A. Verjovsky), World Scientific (1991) 64-167.
In this paper, we give and account of the notion of geometrical finiteness as applied to discrete groups acting on hyperbolic space of any dimension. We prove the equivalence of various definitions of geometrical finiteness, and describe the geometry of fundamental domains. We give a complete description of when Dirichlet domains are finite-sided.
J. Funct. Anal. 113 (1993) 245-317.
The minimal volume of a smooth manifold the infimum of the set of volumes of compatible complete riemannian metrics having sectional curvatures between $ -1 $ and $ 1 $. It was conjectured by Gromov and proved by Bavard and Pansu that the minimal volume of the plane equals $ 2 \pi (1+\sqrt{2}) $. We offer an alternative proof of this result, and describe some related geometric inequalities.
J. Austral. Math. Soc. 55 (1993) 23-40.
We show that a group acting properly discontinuously on a complete simply connected riemannian manifold of pinched negative curvature and fixing an ideal point must be finitely generated. Thus any discrete virtually nilpotent subgroup of isometries of such a manifold is necessarily finitely generated. In the course of the proof, we introduce a notion of rotational part for parabolic isometries.
J. Differential Geom. 38 (1993) 559-583.
Let $ X $ be a complete simply connected riemannian manifold of pinched negative curvature, and let $ X_C $ be the compactification obtained by adjoining the ideal sphere. We show that the closed convex hull of a closed subset of $ X_C $ varies continuously with respect to the Hausdorff topology. We show that the volume of the set of $ n $ points in $ X_C $ is finite, and bounded in terms of $ n $, the dimension of $ X_C $, and the pinching constants of $ X_C $. We also show that this volume varies continuously as a function of these $ n $ points, provided that no two are ever equal to the same ideal point.
Comment. Math. Helv. 69 (1994) 49-81.
We give an example of a subgroup of a discrete cocompact group of isometries of real hyperbolic 4-space which is finitely generated but not finitely presented. The discrete cocompact group can be described as the group generated by reflections in the faces of a right regular 120-cell.
Trans. Amer. Math. Soc. 344 (1994) 391-405.
We describe a notion of geometrical finiteness for groups acting properly discontinuously on complete simply connected manifolds of pinched negative curvature. We prove the equivalence of four different formulations of this notion. We show that any group acting with finite volume quotient is geometrically (hence topologically) finite.
Duke Math. J. 77 (1995) 229-274.
We explore when a complete finite volume hyperbolic 3-manifold fibring over the circle is arithmetic. In the non-compact case, we show that there are only finitely many cyclic commensurability classes of arithmetic hyperbolic surface bundles with any given fibre type. We give a complete classification in the case of once-punctured torus bundles showing that there are precisely three cyclic commensurability classes. We give a partial result for compact manifolds.
Math. Annalen 302 (1995) 31-60.
We give a proof of the result that a path-metric space which satisfies a subquadratic isoperimetric inequality must in fact satisify a linear isoperimetric inquality, and is hence hyperbolic in the sense of Gromov. The argument works for any notion of area satisfying two simple axioms.
Michigan Math. J. 42 (1995) 103-107.
We give an account of complete locally compact locally CAT(1) spaces. We show that a closed geodesic in such a space cannot be freely homotoped to a point through rectifiable curves of length strictly less than $ 2 \pi $. We deduce that such a space is globally CAT(1) if and only if the space of closed curves of length less than $ 2 \pi $ is connected. For these results, we use the Birkhoff curve shortenning process. We give an example of a smooth riemannian metric on the 3-torus for which the Birkhoff curve shortenning process fails to converge. We also describe some area inqualities for surfaces of curvature at most 1.
in ``Geometric group theory'', (ed. R.Charney, M.Davis, M.Shapiro), de Gruyter (1995) 1-48.
Let $ X $ be a complete locally compact simply connected path-metric space which is non-positively curved in the sense of Busemann, namely that the distance function along pairs of geodesic segments is convex. Suppose $ \Gamma $ acts properly discontinuously cocompactly on $ X $. We show that either $ \Gamma $ is hyperbolic or else $ X $ contains a convex subset isometric to a minkowskian plane. If $ X $ is CAT(0), then in the second case, this plane must be euclidean.
Bull. London Math. Soc. 27 (1995) 575-584.
Let $ C $ be the set of simple closed geodesics on a once-punctured torus
with any complete finite-area hyperbolic metric.
McShane's identity says that
$ \sum_{c \in C} {1 \over 1+e^{l(c)}} = {1 \over 2} $
where $ l(c) $ denotes the hyperbolic length of the closed geodesic $ c $.
We translate this into a statement about trees of Markoff triples,
and hence give a new proof of this result.
Bull. London. Math. Soc. 28 (1996) 73-78.
We show how many familiar properties of complete simply connected riemannian manifolds of negative curvature generalise to incomplete manifolds satisfying certain curvature constraints. We show, for example, that such a manifold is geodesically convex, and can be canonically compactified to a topological ball on adjoining the visual sphere. Examples of such manifolds are negatively curved manifolds having bounded geometry up to local rescaling of the metric.
Pacific J. Math. 172 (1996) 1-39.
Let $ M $ be a complete finite-volume hyperbolic 3-manifold
which fibres over the circle with fibre a punctured torus.
Let $ S $ be the set of closed geodesics in $ M $ which are homotopic
to a simple closed curve in the fibre.
We show that
$ \sum_{s \in S} {1 \over 1+e^{l(s)}} = 0 $
where $ l(s) $ denotes the complex length of the closed geodesic $ s $
(i.e. the real hyperbolic length plus $ i $ times the rotational part).
The set $ S $ can be naturally partitioned into two disjoint subsets,
depending on how a curve in $ S $ crosses the stable and unstable foliations
of the fibre.
If we restrict the above sum to one of these subsets, we obtain
(up to sign) the complex modulus of the cusp of $ M $.
These formulae are variations of McShane's identity for a puctured torus,
and are proved here using Markoff triples.
Topology 36 (1997) 325-334.
We study the global behaviour of trees of Markoff triples over the complex numbers. We relate this to the space of type-preserving representations of the once-punctured torus group into $ SL(2,{\Bbb C}) $. In particular, we explore which Markoff triples correspond to quasifuchsian representations. We derive a variation of McShane's identity for quasifuchsian groups. In the case of non-discrete representations, we attempt to relate the asymptotic behaviour of Markoff triples to the realisability of laminations in hyperbolic 3-space. We also consider how some of these issues are related for more general surfaces.
1991 Mathematics Subject Classification: 57M50
Keywords: Markoff triple, quasifuchsian group, punctured torus, representation space, lamination.
Proc. London Math. Soc. 77 (1998) 697-736.
We explore conditions under which the property of geometrical finiteness is open among type-preserving representations of a given group into the group of isometries of hyperbolic $ n $-space. We give general criteria under which this is the case, for example, if every maximal parabolic subgroup has rank at least $ n-2 $. In dimension $ n=3 $, we deduce Marden's theorem that geometrical finiteness is always an open property. We give examples to show that, in general, constraints of the type we describe are necessary in dimension 4 and higher.
Anal. Acad. Sci. Fenn. Math. 23 (1998) 389-414.
We construct continuously many 2-generator groups which all lie in different quasiisometry classes. Unlike the earlier examples of Grigorchuk, the groups can be chosen to be torsion-free and non-amenable.
Comment. Math. Helv. 73 (1998) 232-236.
We give a constuction of the JSJ splitting of a one-ended hyperbolic group, using the separation properties of local cut points in the boundary. We deduce that the JSJ splitting in this case is quasiisometry invariant. We also see that a one-ended non-fuchsian hyperbolic group splits over a two-ended subgroup if and only if it has a local cut point in the boundary. As a corollary, we obtain an annulus theorem for hyperbolic groups. The proofs involve analysing the dynamics of convergence actions on Peano continua.
Acta Math. 180 (1998) 145-186.
We show how particular kinds of group actions on topological trees give rise to isometric actions on $ {\Bbb R} $-trees. As a consequence, one can deduce results about group splittings. For example, we show that if $ \Gamma $ is a finitely presented infinite group such that any ascending chain of finite subgroups eventually stabilises and $ \Gamma $ admits a convergence action on a dendron, then $ \Gamma $ splits over a finite or two-ended subgroup. Using this result one can show that if $ \Gamma $ is a one-ended hyperbolic group with a global cut point in its boundary, then $ \Gamma $ splits over a two-ended subgroup. This is a step towards proving that no such cut point can exist. The paper gives an account of the relation between group actions on trees and foliations of 2-complexes.
Topology 37 (1998) 1275-1298.
We characterise hyperbolic groups as uniform convergence groups. More precisely, we show the following. Suppose $ M $ is a perfect compact metrisable space, and suppose that a group, $ \Gamma $, acts by homeomorphism on $ M $ such that the induced action on the space of disctinct triples of $ M $ is properly discontinuous and cocompact. Then $ \Gamma $ is hyperbolic, and $ M $ is equivariantly homeomorphic to the boundary of $ \Gamma $. We also give a variation on this charactersation, by showing that it is sufficient that $ \Gamma $ should act as a convergence group on $ M $ (i.e. properly discontinuously on the space of distinct triples) such that every point of $ M $ is a conical limit point.
J. Amer. Math. Soc. 11 (1998) 643-667.
We consider splittings of groups over finite and two-ended subgroups. We study the combinatorics of such splittings using generalisations of Whitehead graphs. In the case of hyperbolic groups, we relate this to the topology of the boundary. In particular, we give a proof that the boundary of a one-ended strongly accessible group has no global cut point.
The Epstein Birthday Schrift, Geometry and Topology Monographs Volume 1 (ed. I.Rivin, C.Rourke, C.Series), International Press (1998) 59-97.
We give an account of convergence groups from the point of view of groups which act properly discontinuously on spaces of distinct triples. We give a proof of the equivalence of this characterisation with the dynamical definition of Gehring and Martin. We focus our attention on uniform convergence groups, i.e. those for which the action on the space of distinct triples is also cocompact, and explore some of their properties from a purely dynamical point of view. We show that the space of distinct unordered $ n $-tuples in any continuum is connected. Moreover, the spaces of distinct ordered $ n $-tuples in any metrisable continuum other than a circle or an arc is also connected.
in : Geometric Group Theory Down Under, Proceedings of a Special Year in Geometric Group Theory, Canberra, Australia (ed. J.Cossey, C.F.Miller III, W.D.Neumann, M.Shapiro), de Gruyter (1999) 23-54.
In this paper, we develop the theory of a very general class of treelike structures based on a simple set of betweenness axioms. Within this framework, we explore connections between more familiar treelike objects, such as $ {\Bbb R} $-trees and dendrons. Our principal motive is provide tools for studying convergence actions on continua, and in particular, to investigate how connectedness properties of such continua are reflected in algebraic properties of the groups in question. The main applications we have in mind are to boundaries of hyperbolic and relatively hyperbolic groups and to limit sets of kleinian groups. One of the main results of the present paper constructs dendritic quotients of continua admitting certain kinds of convergence actions, giving us a basis for introducing the techniques of $ {\Bbb R} $-trees into studying such actions. This is a step in showing that the boundary of a one-ended hyperbolic group is locally connected. There are further applications to constructing canonical splittings of such groups, and to limit sets of geometrically finite kleinian groups, which are explored elsewhere. We proceed in a general manner, discussing other connections with $ \Lambda $-trees, protrees, pseudotrees etc. along the way.
1991 Mathematics Subject Classification: 20F32, 20F08.
Key words and phrases: Tree, continuum, cutpoint, convergence group, dendrite, betweenness.
Memoirs Amer. Math. Soc. Volume 662 (1999).
We study convergence group actions on continua, and give a topological criterion which ensures that every global cut point is a parabolic fixed point. As a corollary, we deduce that if G is a relatively hyperbolic group whose boundary is connected, and such that each peripheral subgroup is finitely presented, one-or-two ended and contains no infinite torsion subgroup, then every global cut point of the boundary of G is a parabolic fixed point. We discuss how this relates to other connectedness properties of boundaries.
1991 Mathematics Subject Classification : 20F32.
Trans. Amer. Math. Soc. 351 (1999) 3673-3686.
We show that the limit set of a relatively hyperbolic group with no separating horoball is locally connected if it is connected. On the other hand, if there is a separating horoball centred on a parabolic point, one obtains a non-trivial splitting of the group over a parabolic subgroup relative to the maximal parabolic subgroups. Together with results from elsewhere, one deduces that if $ \Gamma $ is a relatively hyperbolic group such that each maximal parabolic subgroup is one-or-two ended, finitely presented, and contains no infinite torsion subgroup, then the boundary of $ \Gamma $ is locally connected if it is connected. As a corollary, we see that the limit set of a geometrically finite group acting on a complete simply connected manifold of pinched negative curvature must be locally connected if it is connected.
Math. Z. 230 (1999) 509-527.
We describe a variation on the unique product property of groups, which seems natural from a geometric point of view. It is stronger than the unique product property, and hence implies, for example, that the groups rings have no zero divisors. We describe some of its closure properties under extentions and amalgamated free products. We show that most surface groups satisfy this condition, and give various other examples. We explain how these ideas can give a more geometric interpretation of Promislow's example of a non-u.p. group.
J. London Math. Soc. 62 (2000) 813-826.
In this paper we consider group actions on generalised treelike structures (termed ``pretrees'') defined simply in terms of betweenness relations. Using a result of Levitt, we show that if a countable group admits an archimedean action on a median pretree, then it admits an action by isometries on an $ {\Bbb R} $-tree. Thus the theory of isometric actions on $ {\Bbb R} $-trees may be extended to a more general setting where it merges naturally with the theory of right-orderable groups. This approach has application also to the study of convergence group actions on continua.
1991 Mathematics Subject Classification: 20F32
Key words and phrases: tree, archimedean, order, median, pretree, betweenness
Math. Proc. Camb. Phil. Soc. 130 (2001) 383-400.
We define the notion of a ``peripheral splitting'' of a group. This is essentially a representation of the group as the fundamental group of a bipartite graph of groups, where all the vertex groups of one colour are held fixed --- the ``peripheral subgroups''. We develop the theory of such splittings and prove an accessibility result. The main application is to relatively hyperbolic groups with connected boundary, where the peripheral subgroups are precisely the maximal parabolic subgroups. We show that if such a group admits a non-trivial peripheral splitting, then its boundary has a global cut point. Morever, the non-peripheral vertex groups of such a splitting are themselves relatively hyperbolic. These results, together with results from elsewhere, show that under modest constraints on the peripheral subgroups, the boundary of relatively hyperbolic group is locally connected if it is connected. In retrospect, one further deduces that the set of global cut points in such a boundary has a simplicial treelike structure.
2000 Mathematics Subject Classification: 20F67, 20E08.
Trans. Amer. Math. Soc. 353 (2001) 4057-4082.
We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case where all the two-ended subgroups have two-ended commensurators, we obtain an annulus theorem, and a form of JSJ splitting of Rips and Sela. The construction uses ideas from the work of Dunwoody, Sageev and Swenson. We use a particular kind of order structure which combines cyclic orders and treelike structures. In the special case of hyperbolic groups, this provides a link between combinarorial constructions, and constructions arising from the topological structure of the boundary. In this context, we recover the annulus theorem of Scott and Swarup. We also show that a one-ended finitely generated groups which contains an infinite-order element, and such that every infinite cyclic subgroup is (virtually) codimension-one is a virtual surface group.
2000 Subject Classification: 20F65, 20E08.
Trans. Amer. Math. Soc. 354 (2002) 1049-1078.
We describe a variation of the Bergman norm for the algebra of cuts of a connected graph admitting a cofinite group action. By a constuction of Dunwoody, this enables us to obtain nested generating sets for invariant subalgebras. We describe a few applications, in particular, to convergence groups acting on Cantor sets. Under certain finiteness assumptions one can deduce that such actions are necessarily geometrically finite, and hence arise as the boundaries of relatively hyperbolic groups. Similar results have already been obtained by Gerasimov by other methods. One can also use these techniques to give an alternative approach to the Almost Stability Theorem of Dicks and Dunwoody.
2000 Mathematics Subject Classification: 20F65, 20E08.
Pacific J. Math. 207 (2002) 31--60.
We describe a number of characterisations of virtual surface groups which are based on the following result. Let $ G $ be a group and $ k $ be a field. We show that if $ G $ is $ FP_2 $ over $ k $ and if $ H^2(G;k) $, thought of as a $ k $-vector space, contains a 1-dimensional $ G $-invariant subspace, then $ G $ is a virtual surface group (i.e. contains a subgroup of finite index which is the fundamental group of a closed surface other than the sphere or projective plane). In particular, this applies to rational Poincare duality groups. We also conclude that a finitely presented group which is semistable at infinity and with infinite cyclic fundamental group at infinity is a virtual surface group. We recover the result of Mess which characterises such groups as groups which are quasiisometric to complete reimannian planes. We also give a cohomological version of the Seifert conjecture, from which the topological Seifert conjecture (due to Tukia, Mess, Gabai, Casson and Jungreis) can be recovered via work of Zieschang and Scott.
2000 Subject Classification: 20F65, 57M05, 20J06.
J. reine angew. Math. 576 (2004) 11-62.
Let \Gamma be a singly degenerate closed surface group acting properly discontuously on hyperbolic 3-space, {\Bbb H}^3, such that {\Bbb H}^3/\Gamma 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.
2000 Subject Classification: 57M50.
Math. Annalen 332 (2005) 667-676.
We give a brief survey of some recent work on 3-manifolds, notably towards proving Thurston's ending lamination conjecture. We describe some applications to the theory of surfaces and mapping class groups.
in ``European Congress of Mathematics, Stockholm, June 27 -- July 2, 2004'' (ed. A.Laptev) European Mathematical Society Publishing House (2005) 103--115.
We give another proof of the result of Masur and Minsky that the complex of curves associated to a compact orientable surface is hyperbolic.Our proof is more combinatorial in nature and can be expressed mostly in terms of intersection numbers. We show that the hyperbolicity constant is bounded above by a logarithmic function of the complexity of the surface, for example the genus plus the number of boundary components. The geodesics in the complex can be described, up to bounded Hausdorff distance, by a simple relation of intersection numbers.We can also give a similar criterion for recognising if two geodesic segments are a close.
2000 Subject Classification: 20F67, 30F60, 57M50.
J. reine angew. Math. 598 (2006) 105-129
This is a brief introduction to the basic notions of geometric group theory, with particular emphasis on hyperbolic groups and spaces. It is based on a lecture course given at the Tokyo Institute of Technology April-July 2005.
MSJ Mem. Vol 16 (2006).
Let $ \Gamma $ the fundamental group of a compact surface group with non-empty boundary. We suppose that $ \Gamma $ admits a properly discontinuous strictly type preserving action on hyperbolic 3-space such that there is a positive lower bound on the translation lengths of loxodromic elements. We describe the Cannon-Thurston map in this case. In particular, we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3-space, where the action on the circle is obtained by taking any finite-area complete hyperbolic structure on the surface, and lifting to the boundary of hyperbolic 2-space. We deduce that the limit set is locally connected, hence a dentrite in the singly degenerate case. Moreover, we show that the Cannon-Thurson map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3-manifold. For closed surface bundles over the circle, this was obtained by Cannon and Thurston. Some generalisations and variations have been obtained by Minsky, Mitra, Alperin, Dicks, Porti, McMullen and Cannon. We deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected.
2000 Subject Classification: 57M50, 20F67.
Math. Z. 255 (2007) 35-76.
We show that in the game of angel and devil, played on the planar integer lattice, the angel of power 4 can evade the devil. This answers a question of Berlekamp, Conway and Guy. Independent proofs that work for the angel of power 2 have been given by Kloster and by Máthé.
Combin. Probab. Comput. 16 (2007) 245-362.
Let $ M $ be a hyperbolic 3-manifold admitting a homotopy equivalence to a compact surface $ \Sigma $, where the cusps of $ M $ correspond exactly to the boundary components of $ \Sigma $. We construct a nested system of bands in $ M $, where each band is homeomorphic to a subsurface of $ \Sigma $ times an interval. This band system is shown to have various geometrical properties, notably that the boundary of any Margulis tube is mostly contained in the union of the bands. As a consequence, one can deduce the result (conjectured by McMullen and proven by Brock, Canary and Minsky) that the thick part of the convex core of $ M $ has at most polynomial growth. Moreover the degree is at most minus the Euler characteristic of $ \Sigma $. Other applications of this construction to the curve complex of $ \Sigma $ will be discussed elsewhere. The complex is related to the block decomposition of $ M $ described by Minsky, in his work towards Thurston's Ending Lamination Conjecture.
Pacific J. Math. 232 (2007) 1-42.
Let $ M $ be a complete hyperbolic 3-manifold admitting a homotopy equivalence to a compact surface $ \Sigma $, such that the cusps of $ M $ are in bijective correspondence with the boundary components of $ \Sigma $. Suppose we realise a tight geodesic in the curve complex as a sequence of closed geodesics $ M $. There is an upper bound on the lengths of such curves in terms of the lengths of the terminal curves and the topologicial type of $ \Sigma $. We give proofs of these and related bounds. Similar bounds have been proven by Minsky using the sophisticated machinery of hierarchies. Such bounds feature in the work of Brock, Canary and Minsky towards the ending lamination conjecture, and can also be used to study the action of the mapping class group on the curve complex.
Geom. Funct. Anal. 17 (2007) 1001-1042.
The curve graph, $ {\script G} $, associated to a compact surface $ \Sigma $ is the 1-skeleton of the curve complex defined by Harvey. Masur and Minsky showed that this graph is hyperbolic and defined the notion of a tight geodesic therein. We prove some finiteness results for such geodesics. For example, we show that a slice of the union of tight geodesics between any pair of points has cardinality bounded purely in terms of the topological type of $ \Sigma $. We deduce some consequences for the action of the mapping class group on $ {\script G} $. In paricular, we show that it satisfies an acylindricity condition, and that the stable lengths of pseudoanosov elements are rational with bounded denominator.
Invent. Math. 171 (2008) 281-300.
A surface-by-surface group is an extension of a non-trivial orientable closed surface group by another such group. It is an open question as to whether every such group contains a free abelian subgroup of rank 2. We show that, for given base an fibre genera, all but finitely many isomorphism classes of surface-by-surface group contain such an abelian subgroup. This can be rephrased in terms of atoroidal surface bundles over surfaces, or in terms of purely loxodromic surface subgroups of the mapping class groups.
Geom. Funct. Anal. 19 (2009) 943-988.
It has been shown by Bonahon, Agol, Calegari and Gabai that a complete hyperbolic 3-manifold with finitely generated fundamental group is topologically finite. This was conjectured by Marden, and via a geometric reinterpretation by Thurston, has become known as the ``Tameness Theorem''. In this paper we give an account of this result, inspired by arguments of Soma, which is as far as possible self-contained. We also describe how it can be generalised to pinched negative curvature, and how it implies a version of Ahlfors's finiteness theorem in this context.
Enseign. Math. 56 (2010) 229-285.
In this paper, we develop some of the foundations of the theory of relatively hyperbolic groups, as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be ``fine'' if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. This generalises a result of Tukia for geometrically finite kleinian groups.
2000 Subject Classification : 20F57.
Internat. J. Algebra Comput. 22 (2012) 1250016, 66pp.
Let M be a complete hyperbolic 3-manifold with finitely generated fundamental group, and let H be its convex core. We show that there is an upper bound, depending only in the topology of M, on the radius of an embedded hyperbolic ball in H.
Groups, Geom. Dyn. 7 (2013) 109--126
We show that the property of admitting a coarse median structure is preserved under relative hyperbolicity for finitely generated groups.
2010 Mathematics Subject Classification : 20F65
Math. Proc. Camb. Phil. Soc. 154 (2013) 85--95.
We introduce the notion of a coarse median on a metric space. This satisfies the axioms of a median algebra up to bounded distance. The existence of such a median on a geodesic space is quasi-isometry invariant, and so it applies to finitely generated groups via their Cayley graphs. We show that asymptotic cones of such spaces are topological median algebras. We define a notion of rank for a coarse median and show that this bounds the dimension of a quasi-isometrically embedded euclidean plane in the space. Using the centroid construction of Behrstock and Minsky, we show that the mapping class group has this property, and recover the rank theorem of Behrstock and Minsky and of Hamenstadt. We explore various other properties of such spaces, and develop some of the background material regarding median algebras.
2010 Mathematics Subject Classification : 20F65
Pacific J. Math. 261 (2013) 53--93.
In this paper we introduce the notion of a ``stack'' of path-metric spaces. Loosely speaking, this consists of a path-metric space decomposed into a sequence of ``sheets'' indexed by a set of consecutive integers. A stack is a ``hyperbolic'' if it is Gromov hyperbolic and its sheets are uniformly Gromov hyperbolic. We define a Cannon-Thurston map for such a stack, and show that the boundary of a one-sided proper hyperbolic stack is a dendrite. If the stack arises from a sequence compact hyperbolic surfaces with a lower bound on injectivity radius, then this allows us to define an ``ending lamination'' on the surface. We show that the ending lamination has a dynamical property that implies unique ergodicity. We also show that such a sequence is a bounded distance from a Teichm\"uller ray --- a result obtained independently by Mosher. This can be reinterpreted in terms of the Bestvina-Feighn flaring condition, and gives a coarse geometrical characterisation of Teichm\"uller rays. Applying this to a simply degenerate end of a hyperbolic 3-manifold with bounded geometry, we recover this case of Thurston's ending lamination conjecture, proven by Minsky. Various related issues are discussed.
2000 Mathematics Subject Classification : 57M50, 30F60, 20F65.
: in ``Geometry and Topology Down Under'', Contemporary Mathematics Vol. 597, (eds. C.D.Hodgson, W.H.Jaco, M.G.Scharlemann, S.Tillman) American Mathematical Society (2013) 65--138.
We show that a metric median algebra satisfying certain conditions admits a bilipschitz embedding into a finite product of R-trees. This gives rise to a characterisation of closed connected subalgebras of finite products of complete R-trees up to bilipschitz equivalence. Spaces of this sort arise as asymptotic cones of coarse median spaces. This applies to a large class of finitely generated groups, via their Cayley graphs. We show that such groups satisfy the rapid decay property. We also recover the result of Behrstock, Drutu and Sapir, that the asymptotic cone of the mapping class group embeds in a finite product of trees.
2010 Mathematics Subject Classification : 20F65
Geom. Dedicata 170 (2014) 157--176.
We show that there is a universal constant, k, such that the curve graph associated to any compact orientable surface is k-hyperbolic. Independent proofs of this has been given by Aougab and by Clay, Rafi and Schleimer.
2010 Mathematics Subject Classification : 20F65
Pacific J. Math. 269 (2014) 269--280.
Let M by a complete hyperbolic 3-manifold homotopy equivaent to a compact surface Sigma. Let Phi be a subsurface of Sigma, whose boundary is sufficiently short in M. We show that the union of all Margulis tubes homotopic into Phi lifts to a uniformly quasiconvex subset of hyperbolic 3-space.
J. Austral. Math. Soc. 96 (2014) 167--185.
We decribe a number of properties of a median metric space. In particular, we show that a complete connected median metric of finite rank admits a cononical bilipschiz equivalent CAT(0) metric. Metric spaces of this sort arise, up to bilipschitz equivalence, as asymptotic cones of certain classes of finitely generated groups, and the existence of such a structure has various consequences for the large scale geometry of the group.
Groups, Geom. Dyn. 10 (2016) 279--317.
We give conditions under which a quasi-isometric map between direct products of hyperbolic spaces splits as a direct product up to bounded distance and permutation of factors. This is a variation on a result due to Kapovich, Kleiner and Leeb.
Internat. J. Algebra Comput. 26 (2016) 619-633.
We define the strongly separating curve graph to be the full subgraph of the curve graph of a compact orientable surface, where the vertex set consists of all separating curves which do not bound a three-holed sphere. We show that, for all but finitely many surfaces, any automorphism of the strongly separating curve graph is induced by an element of the mapping class group.
Michigan Math. J. 65 (2016) 813-832.
We study the coarse geometry of the Teichmüller space of a compact orientable surface in the Teichmüller metric. We describe when this admits a quasi-isometric embedding of a euclidean space, or a euclidean half-space. We prove quasi-isometric rigidity for Teichmüller space of a surface of complexity at least 2: a result announced independently by Eskin, Rafi and Masur. That is to say, any self-quasi-isometry is a bounded distance from and isometry induced by an element of the mapping class group. We deduce that, apart from some well known coincidences, the Teichmüller spaces are quasi-isometrically distinct. We also show that Teichmüller space satisfies a quadratic isoperimetric inequality. A key ingredient for proving these results is the fact that Teichm\"uller space admits a ternary operation, natural up to bounded distance, which endows the space with the structure of a coarse median space whose rank is equal to the complexity of the surface. From this, one can also deduce that any asymptotic cone is bilipschitz equivalent to a CAT(0) space, and so in particular, is contractible.
J. Topol. 9 (2016) 985-1020.
Suppose we have a finitely presented one-ended group with a purely loxodromic action on a Gromov hyperbolic space satisfying an acylindricity condition. We show that given a generating set, there is an automorphism of the group, and some point in the space which is moved a bounded distance by each of the images of the generators under the automorphism. Here the bound depends only on the group, generating set, and constants of hyperbolicity and acylindricity. With results from elsewhere, this implies that, up to conjugacy, there can only be finitely many purely pseudoanosov subgroups of a mapping class group that are isomorphic to a given finitely presented one-ended group.
In ``Hyperbolic geometry and geometric group theory'' (eds. K.Fujiwara, S.Kojima, K.Ohshika) Advanced Studies in Pure Mathematics, Mathematical Society of Japan (2017) 13-36.
We study the coarse geometry of the mapping class group of a compact orientable surface. We show that, apart from a few low-complexity cases, any quasi-isometric embedding of a mapping class group itself agrees up to bounded distance with a left multiplication. In particular, such a map is a quasi-isometry. This is a strengthening of the result of Hamenstädt and of Behstock, Kleiner, Minsky and Mosher that the mapping class groups are quasi-isometrically rigid. In the course of also develop the general theory of coarse median spaces and median metric spaces with a view to other appications elsewhere.
Pacific J. Math. 293 (2018) 1-73.
Let S be a compact surface, and M be the double of a handlebody. Given a homotopy class of maps from S to M inducing an isomorphism of fundamental groups, we describe a canonical uniformly Lipschitz retraction of the sphere graph of M to the arc graph of S.
J. Topol. Anal. 10 (2018) 245-261.
Let Sigma be a complete finite-area orientable hyperbolic surface with one cusp, and let R be the space of complete geodesic rays in Sigma emanating from the puncture. Then there is a natural action of the mapping class group of Sigma on R. We show that this action is ``almost everywhere'' wandering.
Groups, Geom. Dyn. 12 (2018) 703-719.
These are notes from a mini course on coarse median spaces which I gave that the meeting ``Beyond Hyperbolicity'' in Cambridge in June 2016. The notes were taken by Benjamin Barrett.
In` `Beyond hyperbolicity'', LMS Lecture Note Series, Vol. 454 (ed. M.Hagen, R.C.H.Webb, H.Wilton), Cambridge University Press (2019) 3-24.
We describe a construction which associates to any median metric space a pseudometric satisfying the binary intersection property for closed balls. Under certain conditions, this implies that the resulting space is, in fact, an injective metric space, bilipschitz equivalent to the original metric. In the course of doing this, we derive a few other basic facts about median metrics, and the geometry of CAT(0) cube complexes. One motivation for the study of such metrics is that they arise as asymptotic cones of certain naturally occurring spaces.
Math. Proc. Camb. Phil. Soc. 168 (2020) 43--55.
Let Sigma be a compact orientable surface. Suppose that its fundamental group acts on a Gromov hyperbolic space with hausdorff quotient M. Given any multicurve in Sigma, we can define a shortest realisation in M. Under certain assumptions of the action, we show that such a realisation is supported, up to bounded distance, by a train track realised in M. One purpose of this is to show that certain results in the geometry of hyperbolic 3-manifolds generalise to this context.
September 2010.
We give a generalisation of Minsky's a-priori bounds theorem to Gromov hyperbolic spaces. Let Sigma be a compact orientable surface whose fundamental group acts on a Gromov hyperbolic space. To any curve one can associate its stable length in this action. Suppose we have sequence of simple closed curves that form a tight geodesic in the curve complex of Sigma. Under certain assumptions on the action, we show that one can bound the stable length of each curve in the sequence in terms of the lengths of terminal curves and the topological type of the Sigma. This can be applied to give a description of models of hyperbolic 3-manifolds in term of Gromov hyperbolicity.
September 2010.
We give a characteristion of models of hyperbolic 3-manifolds in terms of Gromov hyperbolicity. Let M be a riemannian manifold with a homotopically equivalence to a closed surface, Sigma, which associates cusps of M to boundary components of Sigma. Under some assumption on M, notably that that universal cover of M is Gromov hyperbolic, we show that the ends of M can be classified as geometrically finite or simply degenerate. To any simply degenerate end we assoicated an ending lamination. We show that any two doubly degenerate manifolds of this sort with the same ending laminations are equivalent, in the sense that their universal covers are equivariantly quasi-isometric. In particular, this applies to constant curvature hyperbolic 3-manifold. A consequence is the ending lamintion conjecture for such manifolds, as proven by Minsky, Brock and Canary.
September 2010.
We give a proof of the general case of the Ending Lamination Theorem, due (in the indecomposable case) to Brock, Canary and Minsky.
First draft: September 2011. Revised: January 2023.
We study the large-scale geometry of Weil-Petersson space, that is, Teichm\"uller space equipped with the Weil-Petersson metric. We show that this admits a natural coarse median structure of a specific rank. Given that this is equal to the maximal dimension of a quasi-isometrically embedded euclidean space, we recover a result of Eskin, Masur and Rafi which gives the coarse rank of the space. We go on to show that, apart from finitely many cases, the Weil-Petersson spaces are quasi-isometrically distinct, and quasi-isometrically rigid. In particular, any quasi-isometry between such spaces is a bounded distance from an isometry. By a theorem of Brock, Weil-Petersson space is equivariantly quasi-isometric to the pants graph, so our results apply equally well to that space.
Preprint, Warwick, March 2015. Revised, January 2020.
Groups, Geom. Dyn. 14 (2020) 607--652.
We consider the limit set of the image of a coarse embedding of a geodesic space into a hyperbolic space of bounded geometry. Under suitable growth conditions on the domain, we show that such a limit set is perfect.
Preprint, July 2017.
We describe the geometry coarse convex hulls in coarse median spaces. When the space comes equipped with a family of projection maps to hyperbolic spaces satisfying certain conditions, we show that coarse hulls are quasi-isometric to CAT(0) cube complexes. From this we deduce a distance formula, and show the existence of monotone paths connecting pairs of points. In particular, we recover a number of results about hierarchically hyperbolic spaces.
Preprint, Warwick, August 2018. Revised, August 2020.
We describe the geometry of a top-dimensional quasiflat (that is, a quasi-isometrically embedded copy of euclidean space of maximal possible dimension) in a coarse median space of finite rank. We show that such a quasiflat is a bounded Hausdorff distance from a finite union of subsets, each of which has a simple structure. In particular, each of these subsets is the image of a direct product of real intervals or rays under a quasi-isometric map which preserves medians up to bounded distance. As one consequence, we recover the result of Behrstock, Hagen and Sisto, regarding quasiflats in asymphoric hierarchically hyperbolic spaces. In the case of a median metric space, one can strenghten the conclusion. In particular, we recover the result of Huang regarding quasiflats in CAT(0) cube complexes, namely that a top-dimensional quasiflat is a finite Hausdorff distance from a finite union of orthants. Examples of coarse median spaces include the mapping class groups of surfaces, as well as Teichmüller space in either the Teichmüller or the Weil-Petersson metric.
Preprint, Warwick, June 2019. Revised, December 2022.
We give an exposition of the Planarity Theorem of Maskit. This gives a classification of finitely generated groups acting effectively properly discontinously by orientation-preserving homeomorphisms on a planar surface. One can also realise such groups as kleinian function groups. We also explain how one can give another proof of the planarity theorem using Dunwoody's theory of tracks.
Preprint, Warwick, June 2019.
Revised, August 2020.
Enseign. Math. 68 (2022) 1-24.
We give a proof that a riemannian manifold of bounded geometry admits a uniformly bilipschitz smooth triangulation. The bilipschitz constants depend only on the curvature and injectivity radius bounds. A number of refinements of this statement, and a generalisation to manifolds with boundary, are described.
April 2020.
We show that a complete simply connected path-metric space which is uniformly locally median is a median metric space. We describe some related results.
First draft, December 2021. Revised, May 2023.
We show that any countable median algebra embeds in a median metric space. If the median algebra has finite rank, one can take the metric space to be compact and connected. A number of related embedding results are given.
First draft: March 2022. Revised: April 2022.
We give a self-contained account of the basic theory of median algebras. We explore notions of betweenness, convexity, walls, duality etc. In this context we include discussion of median metric spaces, cube complexes, spaces with measured walls, coarse median spaces, quasimedian graphs, and related structures.
First draft: March 2022. Revised: March 2024.