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42
The Isomorphism Problem for Toral Relatively Hyperbolic Groups
"... We provide a solution to the isomorphism problem for torsionfree relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually ..."
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Cited by 40 (8 self)
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We provide a solution to the isomorphism problem for torsionfree relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]).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 nmanifolds, for n ≥ 3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsionfree relatively hyperbolic group with abelian parabolics is
Regular neighbourhoods and canonical decompositions for groups
, 2008
"... We find canonical decompositions for finitely presented groups which specialize to the classical JSJdecomposition when restricted to the fundamental groups of Haken manifolds. The decompositions that we obtain are invariant under automorphisms of the group. A crucial new ingredient is the concept o ..."
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Cited by 24 (3 self)
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We find canonical decompositions for finitely presented groups which specialize to the classical JSJdecomposition when restricted to the fundamental groups of Haken manifolds. The decompositions that we obtain are invariant under automorphisms of the group. A crucial new ingredient is the concept of a regular neighbourhood for a finite family of almost invariant subsets of a group. An almost invariant set is an analogue of an immersion.
DEFORMATION SPACES OF TREES
, 2007
"... Let G be a finitely generated group. Two simplicial Gtrees are said to be in the same deformation space if they have the same elliptic subgroups (if H fixes a point in one tree, it also does in the other). Examples include CullerVogtmann’s outer space, and spaces of JSJ decompositions. We discuss ..."
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Cited by 21 (2 self)
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Let G be a finitely generated group. Two simplicial Gtrees are said to be in the same deformation space if they have the same elliptic subgroups (if H fixes a point in one tree, it also does in the other). Examples include CullerVogtmann’s outer space, and spaces of JSJ decompositions. We discuss what features are common to trees in a given deformation space, how to pass from one tree to all other trees in its deformation space, and the topology of deformation spaces. In particular, we prove that all deformation spaces are contractible complexes.
Effective JSJ Decompositions
, 2004
"... In this paper we describe an elimination process which is a deterministic rewriting procedure that on each elementary step transforms one system of equations over free groups into a finitely many new ones. Infinite branches of this process correspond to cyclic splittings of the coordinate group of ..."
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Cited by 19 (4 self)
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In this paper we describe an elimination process which is a deterministic rewriting procedure that on each elementary step transforms one system of equations over free groups into a finitely many new ones. Infinite branches of this process correspond to cyclic splittings of the coordinate group of the initial system of equations. This allows us to construct algorithmically Grushko’s decompositions of finitely generated fully residually free groups and cyclic [abelian] JSJ decompositions of freely indecomposable finitely generated fully residually free groups. We apply these results to obtain an effective description of the set of homomorphisms from a given finitely presented group into a free group, or, more generally, into an NTQ group.
Growth of intersection numbers for free group automorphisms
 J. Topol
"... Abstract. For a fully irreducible automorphism φ of the free group Fk we compute the asymptotics of the intersection number n↦ → i(T, T ′ φ n) for trees T, T ′ in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T ′ φ n for n large. ..."
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Cited by 12 (2 self)
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Abstract. For a fully irreducible automorphism φ of the free group Fk we compute the asymptotics of the intersection number n↦ → i(T, T ′ φ n) for trees T, T ′ in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T ′ φ n for n large.
Boundaries of strongly accessible hyperbolic groups
 GEOMETRY & TOPOLOGY MONOGRAPHS VOLUME 1: THE EPSTEIN BIRTHDAY SCHRIFT PAGES 51–97
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Canonical splittings of groups and 3–manifolds
, 2008
"... We introduce the notion of a ‘canonical’ splitting over Z or Z × Z for a finitely generated group G. We show that when G happens to be the fundamental group of an orientable Haken manifold M with incompressible boundary, then the decomposition of the group naturally obtained from canonical splitting ..."
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Cited by 8 (3 self)
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We introduce the notion of a ‘canonical’ splitting over Z or Z × Z for a finitely generated group G. We show that when G happens to be the fundamental group of an orientable Haken manifold M with incompressible boundary, then the decomposition of the group naturally obtained from canonical splittings is closely related to the one given by the standard JSJdecomposition of M. This leads to a new proof of Johannson’s Deformation Theorem.
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 se ..."
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Cited by 8 (0 self)
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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 quasiisometries, 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