The Isomorphism Problem for All Hyperbolic Groups

Dahmani, François; Guirardel, Vincent

doi:10.1007/s00039-011-0120-0

Cited by 66 publications

(113 citation statements)

References 62 publications

(83 reference statements)

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“…See [Car11] for a related result proved independently by M. Carette, and [Lev05a,DG11] for the one-ended case.…”

Section: Moreover One May Decide Algorithmically Whether Out(g) Is Fmentioning

confidence: 99%

Splittings and automorphisms of relatively hyperbolic groups

Guirardel¹,

Levitt²

2015

Groups Geom. Dyn.

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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 GL n (Z) fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of GL n (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 G) of automorphisms of P that extend to G: it is virtually a McCool group. If Out(P G) is infinite, then P is a vertex group in a splitting of G. If P is torsion-free, then Out(P G) 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 infinitely-ended 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.

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“…See [Car11] for a related result proved independently by M. Carette, and [Lev05a,DG11] for the one-ended case.…”

Section: Moreover One May Decide Algorithmically Whether Out(g) Is Fmentioning

confidence: 99%

Splittings and automorphisms of relatively hyperbolic groups

Guirardel¹,

Levitt²

2015

Groups Geom. Dyn.

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“…Among the three crucial problems for groups raised by Max Dehn in 1911, the isomorphism problem is the most difficult (the word and conjugacy problems constitute the other two: see [13] and [9]). The problem of deciding whether two words (using some algorithms) in some finite generating sets represent the same or conjugate elements in a group constitutes the same word are the word and conjugacy problems respectively.…”

Section: Properties Of the Inverse Graphsmentioning

confidence: 99%

“…Isomorphism problem, on the other hand, has to do with determining whether two groups that appear different are actually the same. In fact, Francois and Vincent asserted that the problem of finding an isomorphism of two groups is unsolvable for some classes of groups, [9]. This is quite interesting as a lot of researches are still going on to solve these problems.…”

Section: Properties Of the Inverse Graphsmentioning

confidence: 99%

Inverse graphs associated with finite groups

Alfuraidan

Zakariya

2017

ejgta

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“…When G is hyperbolic, RC-trees are the Z max -trees of [DG11]: non-trivial edge stabilizers are maximal cyclic subgroups. Proof.…”

Section: Definition 43 a Tree T Is An Rc-tree Ifmentioning

confidence: 99%

“…LetḠ e be the root-closure of G e in G v 1 (hence also in G). As in Section 4.3 of [DG11], we can fold all edges in theḠ e -orbit of e together. Doing this for all edges of T i yields an RC-tree U i which is Mc(H i )-invariant.…”

Section: The Chain Conditionmentioning

confidence: 99%

McCool groups of toral relatively hyperbolic groups

Guirardel

Levitt

2015

Algebr. Geom. Topol.

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The outer automorphism group Out(G) of a group G acts on the set of conjugacy classes of elements of G. McCool proved that the stabilizer Mc(C) of a finite set of conjugacy classes is finitely presented when G is free. More generally, we consider the group Mc(H) of outer automorphisms Φ of G acting trivially on a family of subgroups H i , in the sense that Φ has representatives α i with α i equal to the identity on H i .When G is a toral relatively hyperbolic group, we show that these two definitions lead to the same subgroups of Out(G), which we call "McCool groups" of G. We prove that such McCool groups are of type VF (some finite index subgroup has a finite classifying space). Being of type VF also holds for the group of automorphisms of G preserving a splitting of G over abelian groups.We show that McCool groups satisfy a uniform chain condition: there is a bound, depending only on G, for the length of a strictly decreasing sequence of McCool groups of G. Similarly, fixed subgroups of automorphisms of G satisfy a uniform chain condition.

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The Isomorphism Problem for All Hyperbolic Groups

Cited by 66 publications

References 62 publications

Splittings and automorphisms of relatively hyperbolic groups

Splittings and automorphisms of relatively hyperbolic groups

Inverse graphs associated with finite groups

McCool groups of toral relatively hyperbolic groups

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