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27
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As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually ..."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As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]).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.
Isomorphism problem for finitely generated fully residually free groups
 J. Pure and Applied Algebra
"... Abstract. We prove that the isomorphism problem for finitely generated fully residually free groups (or Fgroups for short) is decidable. We also show that each Fgroup G has a decomposition that is invariant under automorphisms of G, and obtain a structure theorem for the group of outer automorph ..."
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Cited by 23 (2 self)
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Abstract. We prove that the isomorphism problem for finitely generated fully residually free groups (or Fgroups for short) is decidable. We also show that each Fgroup G has a decomposition that is invariant under automorphisms of G, and obtain a structure theorem for the group of outer automorphisms Out(G).
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.
On the automorphism group of generalized BaumslagSolitar groups
"... Abstract. A generalized BaumslagSolitar group (GBS group) is a finitely generated group G which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains nonabelian free groups or is virtually nilpotent of class ≤ 2. It has torsion only at finitely ma ..."
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Cited by 21 (2 self)
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Abstract. A generalized BaumslagSolitar group (GBS group) is a finitely generated group G which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains nonabelian free groups or is virtually nilpotent of class ≤ 2. It has torsion only at finitely many primes. One may decide algorithmically whether Out(G) is virtually nilpotent or not. If it is, one may decide whether it is virtually abelian, or finitely generated. The isomorphism problem is solvable among GBS groups with Out(G) virtually nilpotent. If G is unimodular (virtually Fn × Z), then Out(G) is commensurable with a semidirect product Z k ⋊Out(H) with H virtually free. Contents
CHARACTERIZING RIGID SIMPLICIAL ACTIONS ON TREES
, 2004
"... We extend Forester’s rigidity theorem so as to give a complete characterization of rigid group actions on trees (an action is rigid if it is the only reduced action in its deformation space, in particular it is invariant under automorphisms preserving the set of elliptic subgroups). ..."
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Cited by 12 (2 self)
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We extend Forester’s rigidity theorem so as to give a complete characterization of rigid group actions on trees (an action is rigid if it is the only reduced action in its deformation space, in particular it is invariant under automorphisms preserving the set of elliptic subgroups).
Splittings of generalized Baumslag–Solitar groups
, 2006
"... Abstract. We study the structure of generalized Baumslag–Solitar groups from the point of view of their (usually nonunique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of smallest complexity (fully reduced decompositions) and ..."
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Cited by 11 (1 self)
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Abstract. We study the structure of generalized Baumslag–Solitar groups from the point of view of their (usually nonunique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of smallest complexity (fully reduced decompositions) and give a simplified proof of the existence of deformations. We also prove a finiteness theorem and solve the isomorphism problem for generalized Baumslag–Solitar groups with no nontrivial integral moduli.
Splittings and automorphisms of relatively hyperbolic groups
, 2012
"... We study automorphisms of a relatively hyperbolic group G. When G is oneended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgrou ..."
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Cited by 9 (2 self)
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We study automorphisms of a relatively hyperbolic group G. When G is oneended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GLn(Z) fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of GLn(Z) have to be replaced by McCool groups: automorphisms of parabolic groups acting trivially (i.e. by conjugation) on certain subgroups. Given a malnormal quasiconvex subgroup P of a hyperbolic group G, we view G as hyperbolic relative to P and we apply the previous analysis to describe the group Out(P 1G) of automorphisms of P that extend to G: it is virtually a McCool group. If Out(P 1G) is infinite, then P is a vertex group in a splitting of G. If P is torsionfree, then Out(P 1G) is of type VF, in particular finitely presented. We also determine when Out(G) is infinite, for G relatively hyperbolic. The interesting case is when G is infinitelyended and has torsion. When G is hyperbolic, we show that Out(G) is infinite if and only if G splits over a maximal virtually cyclic subgroup with infinite center. In general we show that infiniteness of Out(G) comes from the existence of a splitting with infinitely many twists, or having a vertex group that is maximal parabolic with infinitely many automorphisms acting trivially on incident edge groups.
A very short proof of Forester’s rigidity result
 GEOM. TOPOL
, 2003
"... The deformation space of a simplicial Gtree T is the set of Gtrees which can be obtained from T by some collapse and expansion moves, or equivalently, which have the same elliptic subgroups as T. We give a short proof of a rigidity result by Forester which gives a sufficient condition for a defo ..."
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Cited by 8 (0 self)
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The deformation space of a simplicial Gtree T is the set of Gtrees which can be obtained from T by some collapse and expansion moves, or equivalently, which have the same elliptic subgroups as T. We give a short proof of a rigidity result by Forester which gives a sufficient condition for a deformation space to contain an Aut(G)invariant Gtree. This gives a sufficient condition for a JSJ splitting to be invariant under automorphisms of G. More precisely, the theorem claims that a deformation space contains at most one strongly slidefree Gtree, where strongly slidefree means the following: whenever two edges e1, e2 incident on a same vertex v are such that Ge1 ⊂ Ge2, then e1 and e2 are in the same orbit under Gv.