Brian H. Bowditch

Cut points and canonical splittings of hyperbolic groups

In this paper, we give a construction of the JSJ splitting of a one-ended hyperbolic group (in the sense of Gromov [Gr]), using the local cut point structure of the boundary. In particular, this gives the quasiisometry invariance of the splitting, as well the annulus theorem for hyperbolic groups. The canonical nature of the splitting is also immediate from this approach. The notion of a JSJ splitting, in this context, was introduced by Sela [Se], who constructed such splittings for all (torsion free) hyperbolic groups. They take their name from the analogy with the characteristic submanifold construction for irreducible 3-manifolds described by Jaco and Shalen [JaS] and Johannson [Jo] (developing a theory outlined earlier by Waldhausen). The JSJ splitting gives a description of the set of all possible splittings of the group over two-ended subgroups, and thus tells us about the structure of the outer automorphism group. We shall take as hypothesis here, the fact that the boundary is locally connected, i.e. a “Peano continuum”. This is now known to be the case for all one-ended hyperbolic groups, from the results of [Bo1,Bo2,L,Sw,Bo5], as we shall discuss shortly. This uses the fact that local connectedness is implied by the non-existence of a global cut point [BeM]. A generalisation of the JSJ splitting to finitely presented groups has been given by Rips and Sela [RS]. The methods of [Se] and [RS] are founded on the theory of actions on R-trees. They consider only splittings over infinite cycic groups. It seems that their methods run into problems if one wants to consider, for example, splittings over infinite dihedral groups (see [MNS]). A more general approach to this has recently been described by Dunwoody and Sageev [DuSa] using tracks on 2-complexes. Fujiwara and Papasoglu have obtained similar results using actions on products of trees [FuP]. These methods work in a more general context than those of this paper. (They deal with splittings of finitely presented groups over “slender” subgroups.) However, one looses some information about the splitting. For example, it is not known if the splitting is quasiisometry invariant in this generality. The annulus theorem would appear to generalise, though this does not follow immediately. A proof of the latter has recently been claimed in [DuSw] for finitely generated groups. (We shall return to this point later.) We shall see that for hyperbolic groups, all these results can be unified in one approach. As we have suggested, deriving the spitting from an analysis of the boundary enables us to conclude that certain topological properties of the boundary are reflected in the structure of the group. For example, we see that the splitting is non-trivial if and only if

Intersection numbers and the hyperbolicity of the curve complex

Abstract 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 close.

A topological characterisation of hyperbolic groups

We characterise word hyperbolic groups as those groups which act properly discontinuously and cocompactly on the space of distinct triples of a compact metrisable space. This is, in turn, equivalent to a convergence group for which every point of the space is a conical limit point.

Connectedness properties of limit sets

We study convergence group actions on continua, and give a criterion which ensures that every global cut point is a parabolic fixed point. We apply this result to the case of boundaries of relatively hyperbolic groups, and consider implications for connectedness properties of such spaces.

Markoff triples and quasifuchsian groups

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 punctured torus group into PSL(2, C). In particular, we explore which Markoff triples correspond to quasifuchsian representations. We derive a variation of McShanes 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 might be related for more general surfaces.

A 4-dimensional Kleinian group

We give an example of a 4-dimensional Kleinian group which is finitely generated but not finitely presented, and is a subgroup of a cocompact Kleinian group.

A Course on Geometric Group Theory

These notes are based on a series of lectures I gave at the Tokyo Institute of Technology from April to July 2005. They constituted a course entitled “An introduction to geometric group theory” totalling about 20 hours. The audience consisted of fourth year students, graduate students as well as several staff members. I therefore tried to present a logically coherent introduction to the subject, tailored to the background of the students, as well as including a number of diversions into more sophisticated applications of these ideas. There are many statements left as exercises. I believe that those essential to the logical developments will be fairly routine. Those related to examples or diversions may be more challenging. The notes assume a basic knowledge of group theory, and metric and topological spaces. We describe some of the fundamental notions of geometric group theory, such as quasi-isometries, and aim for a basic overview of hyperbolic groups. We describe group presentations from first principles. We give an outline description of fundamental groups and covering spaces, sufficient to allow us to illustrate various results with more explicit examples. We also give a crash course on hyperbolic geometry. Again the presentation is rather informal, and aimed at providing a source of examples of hyperbolic groups. This is not logically essential to most of what follows. In principle, the basic theory of hyperbolic groups can be developed with no reference to hyperbolic geometry, but interesting examples would be rather sparse. In order not to interupt the exposition, I have not given references in the main text. We give sources and background material as notes in the final section. I am very grateful for the generous support offered by the Tokyo Insititute of Technology, which allowed me to complete these notes, as well as giving me the freedom to pursue my own research interests. I am indebted to Sadayoshi Kojima for his invitation to spend six months there, and for many interesting conversations. I thank Toshiko Higashi for her constant help in making my stay a very comfortable and enjoyable one. My then PhD student Ken Shackleton accompanied me on my visit, and provided some tutorial assistance. Shigeru Mizushima and Hiroshi Ooyama helped with some matters of translatation etc.

Group actions on trees and dendrons

The main objective of this paper will be to describe how certain results concerning isometric actions on R-trees can be generalised to a wider class of treelike structures. This enables us to analyse convergence actions on dendrons, and hence on more general continua, via the results of [Bo1]. The main applications we have in mind are to boundaries of hyperbolic and relatively hyperbolic groups. For example, in the case of a one-ended hyperbolic group, we shall see that the existence of a global cut point gives a splitting of the group over a two ended subgroup (Corollary 5). Pursuing these ideas further, one can prove the non-existence of global cut points for strongly accessible hyperbolic groups [Bo5], and indeed for hyperbolic groups in general [Sw]. Reset in a general dynamical context, one can use these methods to show, for example, that every global cut point in the limit set of a geometrically finite group is a parabolic fixed point [Bo6]. These results have implications for the algebraic structure of such groups. Some of these are discussed in [Bo3]. The methods of this paper are essentially elementary and self-contained (except of course for the references to what have now become standard results in the theory of R-tree actions). A different approach has been described by Levitt [L2], which gives a generalisation of the central result (Theorem 0.1) to non-nesting actions on R-trees. Levitt’s work makes use of ideas from the theory of codimension-1 foliations (in particular the result of Sacksteder [St]). This seems a more natural result, though our version of Theorem 0.1 suffices for the applications we have in mind at present. There are many further questions concerning the relationship between actions on R-trees and dendrons, which seem worthy of further investigation. Some of these are described in [Bo2]. The notion of an R-tree was formulated in [MoS]. It can be given a number of equivalent definitions. For example, it can be defined simply as a path-metric space which contains no embedded circle. For more discussion of R-trees, see, for example, [Sh,P2]. There is a powerful machinery for studying isometric group actions onR-trees, due to Rips, and generalised by Bestvina and Feighn [BeF] and Gaboriau, Levitt and Paulin [GaLP1]. In most applications, such actions arise from some kind of degeneration of a hyperbolic metric, so that the R-trees obtained come already equipped with a natural metric. However, there are potential applications, as we shall describe, where one obtains, a-priori, only an action by homeomorphism with certain dynamical properties. To deal with this situation, one might attempt either to generalise the Rips machinery to cover such cases, or one might attempt to construct a genuine R-tree starting with a more general action. It is the latter course that is followed in this paper (and that of Levitt

Boundaries of geometrically finite groups

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

Archimedean actions on median pretrees

\Gamma