Hyperbolic structures on manifolds have played a major role in low dimensional topology. A hyperbolic structure is determined by a discrete faithful representation of the fundamental group into the group of isometries of hyperbolic space. In dimensions 2 and 3 the the orientation preserving subgroups are PSL(2,R) and PSL(2,C) respectively. In dimension 2, the applications to complex analysis via Riemann surfaces have long been appreciated, and it remains an active research area. Three dimensional hyperbolic geometry was given a major boost at the end of the 70s by Thurston's geometrisation programme. The aim of this programme is to canonically decompose a 3-manifold into pieces, each of which carries a geometric structure. On a topological level, the canonical decomposition is achieved through the earlier work of Kneser, Milnor, Waldhausen, Johannson, Jaco and Shalen. An important aspect of this is to show that the pieces which "ought to be" hyperbolic really do carry a hyperbolic structure. This was shown by Thurston for the large class of "Haken" 3-manifolds. This work has thrown up a large number of interesting problems in 3-dimensional hyperbolic geometry. Recently a proof of the geometrisation theorem was announced by Perelman.
Starting in dimension 2, the space of finite-area hyperbolic structures on a given surface, known as Teichmuller space, has a very rich structure. A triangulation of the Teichmuller space of a punctured surface has been used by Harer to study cohomological properties of the mapping class group, and is also of interest to physicists. In [1] with David Epstein, we give a geometric construction of such a triangulation using ideas of Andreev as interpreted by Thurston. A related construction for singular euclidean structures is described in [2].
Over the last 25 years much effort has been put into understanding the geometry of ends of hyperbolic 3-manifolds. Notably the tameness conjecture of Marden and ending lamination conjecture of Thurston. There are close connections with the geometry of Teichmuller space, and it is an area of much current interest. In [39]. I give a construction of the Cannon-Thurston map for particular punctured surface groups, adding to work of Minsky, Mitra, Alperin, Dicks, Porti, McMullen and Cannon. This has consequences for the local connectedness of limit sets. In another paper, [50], I discuss ``stacks'' of hyperbolic spaces and set some of this work in a a more general coarse geometrical context. One obtains a coarse characterisation of Teichmuller geodesics in terms of the Bestvina-Feighn flaring condition, and one can recover results of Minsky on Thurston's ending lamination conjecture in the bounded geometry case. Similar results have been obtained independently by Mosher. As another application of these ideas, one can strenghen a result of Miyachi, concerning the limit set of a singly degenerate group with bounded geometry [35].
In the last few years, we have seen proofs of both tameness and the ending lamination conjectures. Significant progress on the former was made by Bonahon, and the project is now complete, through independent work of Agol and of Calegari and Gabai. Some of this has been simplified by Soma. A general account can be found in [45]. The Ending Lamination Conjecture has been a long term-project of Minsky, some of it joint with Masur, and a general resolution has been given through his collaboration with Brock and Canary. An alternative approach this conjecture in [58]. This work is closely related to the geometry of curve complex, mentioned under the heading of ``geometric group theory''. In [41], I give a desciption of a system of bands in hyperbolic 3-manifolds, which has a number of consequences relating to the volume growth of the thick part, as well as to the curve complex. A brief overview of this topic is given in [36]. In [42], I give more direct proofs of certain length bounds of curves in 3-manifolds, closely related to those obtained by Minsky in his work on hierarchies and 3-manifolds. In [43], I use results from the above two papers to derive consequences for the geometry of the acion of the mapping class group in the curve complex. These techniques can also be applied to show that there are only finitely many isomorphism classes of atoroidal surface-by-surface groups [44]. One can, in turn, use these ideas to show that there are only finitely many purely pseudoanosov copies of a given one-ended group in a mapping class group. This has been done by Dahmani and Fujiwara, and independently in [53].
A "Markoff triple" is a triple of complex numbers (x,y,z) satisfying x^2 + y^2 + z^2 = xyz. They arise naturally in number theory, and a connection with 2 and 3-dimensional hyperbolic geometry comes via trace identities in SL(2,R) or SL(2,C). They are discussed in some detail in [18]. In [15], Markoff triples are used to give a proof of McShane's identity concerning hyperbolic structures on the once punctured torus, and in [17], I give a variation of the aforementioned identity to once punctured torus bundles fibring over the circle. (Further variations have been obtained by Sakuma.) They also put in an appearance in [11] (joint with Colin Maclachlan and Alan Reid) where arithmetic 3-manifolds fibring over the circle are studied.
Moving up another dimension, an example of a non-coherent lattice of isometries of hyperbolic 4-space is described in my paper with Geoff Mess [9] (inspired by a construction of M.Kapovich and Potyagailo). "Non-coherent" means that it contains a finitely generated subgroup that is not finitely presented.
In all dimensions the notion of geometrical finiteness introduced by Ahlfors and Greenberg plays a major role. In [5], I study this notion in some detail, and in [19], I consider deformations of geometrically finite representations in higher dimensions.
Many of the constructions of hyperbolic geometry can be adapted the contexts of variable negative or non-positive curvature. A "Hadamard manifold" is a complete simply-connected riemannian manifold of non-positive curvature. Much of my work relates specifically to "pinched Hadamard manifolds" where all the sectional curvatures are assumed to lie between two negative constants. An account of the geometry of Hadamard manifolds and groups acting thereon can be found in the book by Ballmann, Gromov and Schroeder.
In [10], I give an account of geometrical finiteness for groups acting on pinched Hadamard manifolds. This involves showing that discrete parabolic groups in this context are always finitely generated [7]. Recently, Belegradek and V.Kapovitch have gone on the show that the quotient is topologically finite. One aspect of the geometry of hyperbolic space that does not adapt so readily to variable curvature is the construction of convex hulls. Some constructions have been described by Mike Anderson, and in [8] I give some further consideration to this subject. Some features of the geometry of pinched Hadamard manifolds can be generalised to certain classes of incomplete manifolds [16].
It is possible to give a synthetic axiomatisation of upper curvature curvature bounds in a metric space. A theory was developed in the 1940s by Aleksandrov, Toponogov and Busemann. Gromov introduced the term (locally) "CAT(k)" for a space satisfying a comparison axiom intended to capture the notion of a space having curvature bounded above by a constant k. Such spaces have been studied by many authors. A detailed account of CAT(0) spaces can be found in the recent book by Bridson and Haefliger. On a much more modest scale, I give some discussion of CAT(1) spaces in [13]. Vaguely related to the latter is [6], where I discuss the minimal volume of the plane as defined by Gromov and computed by Bavard and Pansu, as well as some related inequalities.
Around 1872, Klein made the proposal of using group theory as a tool to study geometry. Geometric group theory might be viewed as Klein's programme in reverse --- one uses geometric methods to study groups. Two major sources of inspiration in this field are low dimensional topology and hyperbolic geometry. The seminal work of Dehn back in the 1920s can be viewed in this context, as can the subsequent development of "small cancellation theory". The work of Thurston in the late 70s which intimately linked 3-manifold theory and hyperbolic geometry gave a major boost to this subject, and it is has now grown into a major field in its own right. A short introduction to geometric group theory is given in [38],
A great deal of the pioneering work in this area has been carried out by Gromov. Much of my recent work relates to his notions of hyperbolic, or relatively hyperbolic, groups. These are groups which are "negatively curved" on a large scale. Examples of hyperbolic groups are fundamental groups of compact negatively curved manifolds. If we allow for finite volume manifolds, then we get examples of relatively hyperbolic groups. An account (among several) of Gromov's hyperbolicity hypothesis can be found in [4] and an account of relatively hyperbolic groups is given in [46]. Canonically associated to a (relatively) hyperbolic group is its boundary, with is a compact metrisable space. The group acts on its boundary as a convergence group, in the sense defined by Gehring and Martin, and elaborated upon by Tukia and Freden. Some discussion is given in [25]. In fact, a hyperbolic group can be characterised as a group which admits a ``uniform'' convergence action on a compact metrisable space [23] (that is, a convergence action for which every point is a conical limit point). This characterisation has been generalised to relatively hyperbolic groups by Yaman. It was noted by Gromov that a hyperbolic group can be characterised as one having a subquadratic Dehn function. More detailed proofs have been given by Delzant, Coornaert and Papadopoulos, Olshanskii, Papasoglu and by me [12]. Another observation of Gromov is that if a CAT(0) space admits a discrete cocompact action, then either it contains an embedded euclidean plane, or else the group is hyperbolic. Detailed proofs have been supplied by Heber, Bridson and (in a slightly more general setting) by me [14]. An interesting open question asks whether, in the former case, the plane can be chosen to be periodic.
The Bass-Serre theory of groups splittings has received a great boost from 3-manifold ideas, in particular through work of Stallings and Dunwoody. More recently, the "characteristic submanifold" construction of Waldhausen, Johansson, Jaco and Shalen has found an analogue in the "JSJ" splittings of groups, initially due to Sela, and further developed by Rips, Dunwoody, Sageev, Fujiwara, Papasoglu etc. An important technique in this area is the theory of group actions on R-trees, introduced by Morgan and Shalen, and further developed by Rips, Bestvina, Feighn, Gaboriau, Levitt, Paulin etc.
In [26], I develop a theory of very general treelike structures and group actions upon them, with applications to convergence actions on continua. (Similar structures have been studied by Ward, and more recently by Adeleke and Neumann.) In [22], I show how one can construct actions on R-trees from certain kinds of topological actions on trees. This is elaborated on in [30] with John Crisp, using a result of Levitt.
The main motivation of the above work was to analyse properties of (relatively) hyperbolic groups from the topological properties of their boundaries, and the dynamics of the convergence group actions on them. One important consequence, using results of Levitt and Swarup is that the boundary of a hyperbolic group has no global cutpoint --- see [27] and [24]. It now follows by a result of Bestvina and Mess that such a boundary is locally connected, i.e. a Peano continuum. This greatly simplifies the study of such boundaries. In particular, one arrives at a pure topological construction of the JSJ splitting of a hyperbolic group, as well as the annulus theorem of Scott and Swarup (generalised by Dunwoody and Swenson) [21]. One can tie in these topological constructions with combinatorial constructions that work in the general context of finitely generated groups [32].
One can carry over much of this analysis to relatively hyperbolic groups. If the boundary of such a group is connected then (under some mild hypothesis on the peripheral subgroups) every global cut point is a parabolic fixed point. One can go on to show that the set of global cut points gives rise to a splitting of the group over subgroups of peripheral subgroups [31]. Using this one can deduce that such boundaries are locally connected [28]. In particular, the limit of a geometrically finite kleinian group, in any dimension, is locally connected if it connected. The same result applies to groups acting on pinched Hadamard manifolds. In another direction, we study convergence group actions on Cantor sets in [33], showing that such groups are relatively hyperbolic under certain finiteness assumptions. Similar results have been obtained by Gerasimov.
In 1981, Harvey introduced the "curve complex" associated to a compact surface. It has since been used by Harer and Ivanov to study the mapping class group. It is related to the Harer complex refered to earlier. Recently, Masur and Minsky showed that this complex is hyperbolic in the sense of Gromov. This forms part of the work of Minsky and his collaborators towards the ending lamination conjecture, also referred to above. In [37], we give another proof of Masur and Minsky's result. In fact, one can show that the curve graph (its 1-skeleton) is uniformly hyperbolic, in the sense that the hyperbolicity constant does not depend on the topolical type of the surface. This has been shown independenty by Aougab, by Clay, Rafi and Scheimer, by Hensel, Przytycki and Webb, and in [52]. A result about projecting sphere graphs to arc graphs is described in joint work with Francesca Iezzi [66], and gives a different proof of a result of Hensel and Hamenstadt.
Technology developed towards solving the Ending Lamination conjecture, in particular, variations on the curve complex, and ``subsurface projections'' between them, has found widespread applications elsewhere. Notable among these has been the study of the large scale geometry of the mapping class groups, and Teichmuller space. For example, it has been shown by Berhstock, Kleiner, Minksy and Mosher, and independently by Hamenstadt, that the mapping class group is quasi-isometricallly rigid: that is, any quasi-isometry is a bounded distance from an isometry. Another proof of rigidy for the mapping class group is given in [62] which also shows that self-quasi-isometric embeddings are necessarily quasi-isometries. In [64] it is shown that Teichmuller space in the Teichmuller metric is also quasi-isometrically rigid --- a result obtained independently by Eskin, Masur and Rafi by different methods. In [63] it is shown that Teichmuller space in the Weil-Peterson metric is also quasi-isometrically rigid, apart from finitely many surface types. This last result uses a combinatorial rigidity result for the ``strongly separating'' curve graphs, proven in [61] Some related statements from products of hyperbolic spaces are described in [60].
The above makes use of a notion of ``coarse median space''. This is a geodesic metric space which admits a ternary operation, which in a certain sence, behaves like a median algebra up to bounded distance. These are introduced in [49] where various consequences are given. For example, one can bound the coarse rank of such a space, that is the maximal dimension of a quasi-isometrically embedded eucliean space. The fact that the mapping class group admits such a structure follows from a constuction of Behrstock and Minsky. One can also show that various other spaces, such as Teichmuller space, admit such a stucture. Moreover, it is shown in [48] that is invariant under relative hyperbolicity. To apply this property, one shows that (under certain hypotheses) the asymptotic cone of such a space embeds in a finite product of R-trees [51] From this it follows that it is bilipschitz equivalent to a median metric space. It in turn follows that it is bilipschitz equivelent to a CAT(0) metic, [54], and also to an injective metric [65].
In [34], I describe some characterisations of planar groups, i.e. groups that are virtually the fundamental groups of closed surfaces (other than the sphere or projective plane). In particular, planar groups are precisely 2-dimensional rational Poincare duality groups. Moreover, one gets a cohomological version of the Seifert conjecture as conjectured by Scott. One also recovers Mess's characterisation of planar groups as groups that are quasiisometric to complete riemannian planes.
In some other directions, I describe a geometric approach to the zero-divisor problem for certain groups in [29], and give a means of constructing uncountably many quasiisometry classes of finitely generated groups in [20].
Last but not least are two papers which don't fall readily into any of the above categories. In [3], together with Paterson and McColl, we describe patterns of arcs in the disc, motivated by a question in VLSI circuits. In [40], I give a solution to the angel problem of Conway, Berlekamp and Guy. Independent solutions have been given by Kloster, by Mathe and by Gacs.