Daniel Groves

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special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]).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.

Limit groups for relatively hyperbolic groups. I. The basic tools

We begin the investigation of -limit groups, where is a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of (16), we adapt the re- sults from (21) and (22) to this context. Specifically, given a finitely generated group G, and a sequence of pairwise non-conjugate ho- momorphisms {hn : G ! }, we extract an R-tree with a nontrivial isometric G-action. This, along with the analogue of Selas shortening argument allows us to prove the main result of this paper, that is Hopfian. In his remarkable series of papers (38, 39, 41), Z. Sela has classified those finitely generated groups with the same elementary theory as the free group of rank 2 (see also (40) for a summary). This class includes all nonabelian free groups, most surface groups, and certain other hyperbolic groups. In particular, Sela answers in the positive some long-standing questions of Tarski (Kharlampovich and Miasnikov have another approach to these problems; see (29)). In (38), Sela begins with a study of limit groups. Selas definition of a limit group is geometric, though it turns out that a group is a limit group if and only if it is a finitely generated fully-residually free group. Sela then produces Makanin-Razborov diagrams, which give a parametrization of Hom(G,F), where G is an arbitrary finitely gener- ated group and F is a nonabelian free group (such a parametrisation is also given in (28)). Over the course of his six papers, two of the main tools Sela uses are the theory of isometric actions on R-trees and the shortening argument. Selas work naturally raises the question of which other classes of groups can be understood using Selas approach. Many of Selas meth- ods (and, more strikingly, some of the answers) come from geometric group theory. Thus it seems natural to consider, when looking for classes of groups to apply these methods to, groups of interest in geo- metric group theory. In (42), Sela considers an arbitrary torsion-free

Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams

Let be a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. We construct Makanin–Razborov diagrams for . We also prove that every system of equations over is equivalent to a finite subsystem, and a number of structural results about –limit groups.

Residual finiteness, QCERF and fillings of hyperbolic groups

We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling. 20E26, 20F67, 20F65 A group G is residually finite (or RF) if for every g2 GXf1g, there is some finite group F and an epimorphism W G! F so that . g/⁄ 1. In more sophisticated language G is RF if and only if the trivial subgroup is closed in the profinite topology on G . If H < G , then H is separable if for every g2 GXH , there is some finite group F and an epimorphism W G! F so that . g/O. H/. Equivalently, the subgroup H is separable in G if it is closed in the profinite topology on G . If every finitely generated subgroup of G is separable, G is said to be LERF or subgroup separable. If G is hyperbolic, and every quasi-convex subgroup of G is separable, we say that G is QCERF. In this paper, we show that if every hyperbolic group is RF, then every hyperbolic group is QCERF.

Enumerating limit groups

We prove that the set of limit groups is recursively enumerable, answering a question of Delzant. One ingredient of the proof is the observation that a finitely presented group with local retractions (a la Long and Reid) is coherent and, furthermore, there exists an algorithm that computes presentations for finitely generated subgroups. The other main ingredient is the ability to algorithmically calculate centralizers in relatively hyperbolic groups. Applications include the existence of recognition algorithms for limit groups and free groups.

Conjugacy classes of solutions to equations and inequations over hyperbolic groups

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results to enumerate the immutable subgroups of a torsion-free hyperbolic group.

Detecting free splittings in relatively hyperbolic groups

We describe an algorithm which determines whether or not a group which is hyperbolic relative to abelian groups admits a nontrivial splitting over a finite group.

Limits of (certain) CAT(0) groups, I: Compactification

The purpose of this paper is to investigate torsion-free groups which act properly and cocompactly on CAT(0) metric spaces which have isolated flats, as defined by Hruska [18]. Our approach is to seek results analogous to those of Sela, Kharlampovich and Miasnikov for free groups and to those of Sela (and Rips and Sela) for torsion-free hyperbolic groups. This paper is the first in a series. In this paper we extract an R-tree from an asymptotic cone of certain CAT(0) spaces. This is analogous to a construction of Paulin, and allows a great deal of algebraic information to be inferred, most of which is left to future work.

COFINITELY HOPFIAN GROUPS, OPEN MAPPINGS AND KNOT COMPLEMENTS

M.R. BRIDSON, D. GROVES, J.A. HILLMAN, AND G.J. MARTINAbstract. A group Γ is defined to be cofinitely Hopfian if everyhomomorphism Γ → Γ whose image is of finite index is an auto-morphism. Geometrically significant groups enjoying this propertyinclude certain relatively hyperbolic groups and many lattices. Aknot group is cofinitely Hopfian if and only if the knot is not atorus knot. A free-by-cyclic group is cofinitely Hopfian if and onlyif it has trivial centre. Applications to the theory of open mappingsbetween manifolds are presented.

Fillings, finite generation and direct limits of relatively hyperbolic groups

We examine the relationship between finitely and infinitely generated relatively hyperbolic groups. We observe that direct limits of relatively hyperbolic groups are in fact direct limits of finitely generated relatively hyperbolic groups. We combine this observation with known results to prove the Strong Novikov Conjecture for some exotic groups constructed by Osin.