Normal automorphisms of relatively hyperbolic groups

Abstract. We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNNextensions, one-relator groups, automorphism groups of polynomial algebras, 3-manifold groups and graph products. Acylindrical hyperbolicity is then used to obtain some results about the algebraic structure, analytic properties and measure equivalence rigidity of groups from these classes.

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“…The proof is based on the approach suggested in [65] and essentially uses acylindrical hyperbolicity.…”

Section: Corollary 210 a Group G ∈ M 3 Is Inner Amenable Iff It Is mentioning

confidence: 99%

Acylindrical hyperbolicity of groups acting on trees

2014

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“…Let Lab .q 1 / 1 Á R 1 T 1 R 2 T 2 be the corresponding decomposition of the label. By (20) and (21), we have (19) and the triangle inequality, we obtain…”

Section: Torsion In the Quotientmentioning

confidence: 98%

“…In the present paper we apply small cancellations over relatively hyperbolic groups to prove embedding theorems for countable groups. Further applications of our methods can be found in [2], [1], [4], [19], [20], [25].…”

Section: Introductionmentioning

confidence: 99%

Small cancellations over relatively hyperbolic groups and embedding theorems

2010

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We generalize the small cancellation theory over ordinary hyperbolic groups to relatively hyperbolic settings. This generalization is then used to prove various embedding theorems for countable groups. For instance, we show that any countable torsion free group can be embedded into a finitely generated group with exactly two conjugacy classes. In particular, this gives the affirmative answer to the wellknown question of the existence of a finitely generated group G other than ‫ޚ2=ޚ‬ such that all nontrivial elements of G are conjugate.

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“…We may assume that there is a non-trivial h ∈ H such that α(h) = h. If H is abelian, malnormality of maximal abelian subgroups implies that α is the identity on H. If not, the result follows from Lemma 5.2 of [MO10] (which is valid for any homomorphism ϕ : H → G, not just automorphisms of H), see also Corollary 7.4 of [AMS13].…”

Section: Proof Of the Other Resultsmentioning

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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Normal automorphisms of relatively hyperbolic groups

Cited by 36 publications

References 40 publications

Acylindrical hyperbolicity of groups acting on trees

Acylindrical hyperbolicity of groups acting on trees

Small cancellations over relatively hyperbolic groups and embedding theorems

McCool groups of toral relatively hyperbolic groups

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