Height in splittings of hyperbolic groups

SupposeH is a hyperbolic subgroup of a hyperbolic groupG. Assume there existsn > 0 such that the intersection ofn essentially distinct conjugates ofH is always finite. Further assumeG splits overH with hyperbolic vertex and edge groups and the two inclusions ofH are quasi-isometric embeddings. ThenH is quasiconvex inG. This answers a question of Swarup and provides a partial converse to the main theorem of [23].     

Vertex finiteness for splittings of relatively hyperbolic groups

Consider a group G and a family A of subgroups of G. We say that vertex finiteness holds for splittings of G over A if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in A.     

Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut-points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well-known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with infinite groups of outer automorphisms, and co-Hopfian relatively hyperbolic groups.     

Growth of relatively hyperbolic groups

We show that a finitely generated group that is hyperbolic relative to a collection of proper subgroups either is virtually cyclic or has uniform exponential growth.

     

Dehn filling in relatively hyperbolic groups

We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, preferred paths, is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a group-theoretic analog of the Gromov-Thurston 2π Theorem in the context of relatively hyperbolic groups. The first author was supported in part by NSF Grant DMS-0504251. The second author was supported in part by an NSF Mathematical Sciences Post-doctoral Research Fellowship. Both authors thank the NSF for their support. Most of this work was done while both authors were Taussky-Todd Fellows at Caltech.     

Boundary amenability of relatively hyperbolic groups

Let K be a fine hyperbolic graph and Γ be a group acting on K with finite quotient. We prove that Γ is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. We prove this by showing that the group Γ acts amenably on a compact topological space. We include some applications to the theories of group von Neumann algebras and of measurable orbit equivalence relations.     

Limit sets of relatively hyperbolic groups

In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known results on limit sets of geometrically finite Kleinian groups are derived in general convergence groups. We also establish dynamical quasiconvexity of undistorted subgroups in finitely generated groups with nontrivial Floyd boundaries.     

Peripheral fillings of relatively hyperbolic groups

In this paper a group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group G we define a peripheral filling procedure, which produces quotients of G by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3-manifold M on the fundamental group π₁(M). The main result of the paper is an algebraic counterpart of Thurston’s hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of G ‘almost’ have the Congruence Extension Property and the group G is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings. Mathematics Subject Classification (2000) 20F65, 20F67, 20F06, 57M27, 20E26     

Floyd maps for relatively hyperbolic groups

Let ${\mathbf{delta}_{\mathcal S,\lambda}}$ denote the Floyd metric on a discrete group G generated by a finite set ${\mathcal S}$ with respect to the scaling function f _n ?= λ ⁿ for a positive λ <?1. We prove that if G is relatively hyperbolic with respect to a collection ${\mathcal P}$ of subgroups then there exists λ such that the identity map ${G\to G}$ extends to a continuous equivariant map from the completion with respect to ${\mathbf{\delta}_{\mathcal S,\lambda}}$ to the Bowditch completion of G with respect to ${\mathcal P}$ . In order to optimize the proof and the usage of the map theorem we propose two new definitions of relative hyperbolicity equivalent to the other known definitions. In our approach some “visibility” conditions in graphs are essential. We introduce a class of “visibility actions” that contains the class of relatively hyperbolic actions. The convergence property still holds for the visibility actions. Let a locally compact group G act on a compactum Λ with convergence property and on a locally compact Hausdorff space Ω properly and cocomactly. Then the topologies on Λ and Ω extend uniquely to a topology on the direct union ${T=\Lambda{\sqcup}\Omega}$ making T a compact Hausdorff space such that the action ${G{\curvearrowright}T}$ has convergence property. We call T the attractor sum of Λ and Ω.     

Accidental parabolics and relatively hyperbolic groups

By constructing, in the relative case, objects analogous to Rips and Sela’s canonical representatives, we prove that the set of conjugacy classes of images by morphisms without accidental parabolic, of a finitely presented group in a relatively hyperbolic group, is finite.     

Conformal dimension and canonical splittings of hyperbolic groups

We prove a general criterion for a metric space to have conformal dimension one. The conditions are stated in terms of the existence of enough local cut points in the space. We then apply this criterion to the boundaries of hyperbolic groups and show an interesting relationship between conformal dimension and some canonical splittings of the group.     

Aspherical manifolds with relatively hyperbolic fundamental groups

We show that the aspherical manifolds produced via the relative strict hyperboli- zation of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automorphism group, absence of splitting over elementary subgroups, and acylindricity. In fact, some of these properties hold for any compact aspherical manifold with incompressible aspherical boundary components, provided the fundamental group is hyperbolic relative to fundamental groups of boundary components. We also show that no manifold obtained via the relative strict hyperbolization can be embedded into a compact Kähler manifold of the same dimension, except when the dimension is two.     

Existential questions in (relatively) hyperbolic groups

We study the decidability of the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. Our tools are Rips and Sela’s canonical representatives for these groups, and solvability of equations with rational constraints (involving finite state automata) in free groups and free products.     

On quasiconvexity and relatively hyperbolic structures on groups

Let G be a group which is hyperbolic relative to a collection of subgroups H₁{\mathcal{H}_1}, and it is also hyperbolic relative to a collection of subgroups H₂{\mathcal{H}_2}. Suppose that H₁ ì H₂{\mathcal{H}_1 \subset \mathcal{H}_2}. We characterize when a relative quasiconvex subgroup of (G, H₂){(G, \mathcal H_2)} is still relatively quasiconvex in (G, H₁){(G, \mathcal H_1)}. We also show that relative quasiconvexity is preserved when passing from (G, H₁){(G, \mathcal H_1)} to (G, H₂){(G, \mathcal H_2)}. Applications are discussed.     

The isomorphism problem for toral relatively hyperbolic groups

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for n≥3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabolics is algorithmically constructible.     

Divergence spectra and Morse boundaries of relatively hyperbolic groups

We introduce a new quasi-isometry invariant, called the divergence spectrum, to study finitely generated groups. We compare the concept of divergence spectrum with the other classical notions of divergence and we examine the divergence spectra of relatively hyperbolic groups. We show the existence of an infinite collection of right-angled Coxeter groups which all have exponential divergence but they all have different divergence spectra. We also study Morse boundaries of relatively hyperbolic groups and examine their connection with Bowditch boundaries.     

On the residual finiteness of outer automorphisms of relatively hyperbolic groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.     

The isocohomological property, higher Dehn functions, and relatively hyperbolic groups

The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type HF^∞, i.e. that has a classifying space with the homotopy type of a polyhedral complex with finitely many cells in each dimension, we show that the isocohomological property is geometric and is equivalent to the property that the universal cover of the classifying space has polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable, respectively HF^∞ and isocohomological, then we show that the group itself has these respective properties. Combining with the results of Connes-Moscovici and Dru?u-Sapir we conclude that a group satisfies the strong Novikov conjecture if it is hyperbolic relative to subgroups which are of property RD, of type HF^∞ and isocohomological.     

Peripheral splittings of groups

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 theory mainly applies 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. Moreover, 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 a 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.