Vertex finiteness for splittings of relatively hyperbolic groups

Guirardel, Vincent; Levitt, Gilbert

doi:10.1007/s11856-016-1304-x

Cited by 2 publications

(3 citation statements)

References 28 publications

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“…The Klein bottle group is 2-CSA but not 1-CSA. Any hyperbolic group Γ is K-CSA for some K since finite subgroups of Γ have bounded order, and there are only finitely many isomorphism classes of virtually cyclic groups whose finite subgroups have bounded order (see Lemma 2.2 of [GL16] for a proof). Corollary 9.10 will say that Γ-limit groups also are K-CSA.…”

Section: Csa Groupsmentioning

confidence: 99%

JSJ decompositions of groups

Guirardel,

Levitt

2016

Preprint

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This is an account of the theory of JSJ decompositions of finitely generated groups, as developed in the last twenty years or so.We give a simple general definition of JSJ decompositions (or rather of their Bass-Serre trees), as maximal universally elliptic trees. In general, there is no preferred JSJ decomposition, and the right object to consider is the whole set of JSJ decompositions, which forms a contractible space: the JSJ deformation space (analogous to Outer Space).We prove that JSJ decompositions exist for any finitely presented group, without any assumption on edge groups. When edge groups are slender, we describe flexible vertices of JSJ decompositions as quadratically hanging extensions of 2-orbifold groups.Similar results hold in the presence of acylindricity, in particular for splittings of torsion-free CSA groups over abelian groups, and splittings of relatively hyperbolic groups over virtually cyclic or parabolic subgroups. Using trees of cylinders, we obtain canonical JSJ trees (which are invariant under automorphisms).We introduce a variant in which the property of being universally elliptic is replaced by the more restrictive and rigid property of being universally compatible. This yields a canonical compatibility JSJ tree, not just a deformation space. We show that it exists for any finitely presented group.We give many examples, and we work throughout with relative decompositions (restricting to trees where certain subgroups are elliptic).

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Section: Csa Groupsmentioning

confidence: 99%

JSJ decompositions of groups

Guirardel,

Levitt

2016

Preprint

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“…Lemma 2.3 then implies that ρ v (Mc 0 (H i )) is a finite subgroup of G v , but we need to bound its order only in terms of G (independently of the sequence H i ). To get this uniform bound, we note that there are only finitely many possibilities for G v up to isomorphism by [GL13]. Moreover Out(G v ) is virtually torsion-free by [GL12, Cor 4.5], so there is a bound for the order of its finite subgroups.…”

Section: The Index Of Outmentioning

confidence: 99%

“…The group G v i is then isomorphic to the fundamental group of a compact surface Σ i , and incident edge groups are boundary subgroups. The topology of Σ i may vary with i, but the number of boundary components of Σ i is bounded (by a simple accessibility argument, or because the rank of G v i as a free group is bounded by [GL13]). …”

Section: The Index Of Outmentioning

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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Vertex finiteness for splittings of relatively hyperbolic groups

Cited by 2 publications

References 28 publications

JSJ decompositions of groups

JSJ decompositions of groups

McCool groups of toral relatively hyperbolic groups

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