Commensurating endomorphisms of acylindrically hyperbolic groups and applications

Antolín, Yago; Minasyan, Ashot; Sisto, Alessandro

doi:10.4171/ggd/379

Cited by 35 publications

(58 citation statements)

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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: 93%

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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Section: Proof Of the Other Resultsmentioning

confidence: 93%

McCool groups of toral relatively hyperbolic groups

Guirardel

Levitt

2015

Algebr. Geom. Topol.

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“…We fix an element g ∈ G n+1 . Since tr B n (g)) 1 1, for every B ∈ B we have ϕ B (tr B n (g)) V ϕ ∞ . Moreover, since ϕ B is alternating, by Lemma 4.6 there are at most n(n + 1) cosets B ∈ B such that ϕ B (tr B n (g)) = 0, so…”

Section: Proof For Every Coset B ∈ B We Define a Cochain ϕmentioning

confidence: 99%

Extending higher-dimensional quasi-cocycles

2015

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Abstract. Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(Fn) for n ≥ 2). We prove that, in degree 3, the bounded cohomology of G with real coefficients is infinite-dimensional. Our proof is based on an extension to higher degrees of a recent result by Hull and Osin. Namely, we prove that, if H is a hyperbolically embedded subgroup of G and V is any R[G]-module, then any n-quasi cocycle on H with values in V may be extended to G. Also, we show that our extensions detect the geometry of the embedding of hyperbolically embedded subgroups, in a suitable sense.Bounded cohomology of discrete groups is very hard to compute. For example, as observed in [Mon06], there is not a single countable group G whose bounded cohomology (with trivial coefficients) is known in every degree, unless it is known to vanish in all positive degrees (this is the case, for example, of amenable groups). Even worse, no group G is known for which the supremum of the degrees n such that H n b (G, R) = 0 is positive and finite.In degree 2, bounded cohomology has been extensively studied via the analysis of quasi-morphisms (see e.g. [Bro81, EF97, Fuj98, BF02, Fuj00] for the case of trivial coefficients, and [HO13, BBF13] for more general coefficient modules). In this paper we exploit quasi-cocycles, which are the higherdimensional analogue of quasi-morphisms, to prove non-vanishing results for bounded cohomology in degree 3. To this aim, we prove that quasi-cocycles may be extended from a hyperbolically embedded family of subgroups to the ambient group. We also discuss the geometric information carried by our extensions of quasi-cocycles, showing in particular that projections on hyperbolically embedded subgroups may be reconstructed from the extensions of suitably chosen quasi-cocycles.Quasi-cocycles and bounded cohomology. Let G be a group, and V be a normed R[G]-space, i.e. a normed real vector space endowed with an

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“…Thus we can conclude that

{normalE}_{F} false(N false) = {normalE}_{F} false(F false)

, but

{normalE}_{F} false(F false) = false{1 false}

F

contains no non‐trivial finite normal subgroups, therefore

{normalE}_{F} false(N false) = false{1 false}

. We can now apply [, Lemma 5.12] to find a loxodromic element

h \in N

such that

{normalE}_{F} false(h false) = false⟨ h false⟩ {normalE}_{F} false(N false) = false⟨ h false⟩

. Since

{normalC}_{F} false(h false) \subseteq {normalE}_{F} false(h false)

by , we deduce that

{normalC}_{F} false(h false) = false⟨ h false⟩

, as required.…”

Section: Introductionmentioning

confidence: 99%

“…Now, if

F

acts on a

δ

‐hyperbolic metric space

scriptS

co‐boundedly and

H ⩽ F

is a non‐elementary subgroup containing at least one loxodromic element then, by [, Lemma 5.6], there is a largest finite subgroup

{normalE}_{F} false(H false) ⩽ F

, normalized by

H

. In particular,

F

itself has a maximal finite normal subgroup

{normalE}_{F} false(F false)

(cf.…”

Section: Introductionmentioning

confidence: 99%

“…Since

{normalC}_{F} false(h false) \subseteq {normalE}_{F} false(h false)

by , we deduce that

{normalC}_{F} false(h false) = false⟨ h false⟩

, as required. If

x \notin false⟨ h false⟩ = {normalE}_{F} false(h false)

, then the final claim of the lemma follows from [, Lemma 5.13]. Otherwise, if

x = h^{m}

for some

m \in Z

, then we choose

n = 1 - m \in Z

, so that

f = h^{n} x = h

, and the required equality

{normalC}_{F} false(f false) = {normalE}_{F} false(f false) = false⟨ f false⟩

holds for

f

because it holds for

h

□

…”

Section: Introductionmentioning

confidence: 99%

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On conjugacy separability of fibre products

Minasyan

2017

Proc. London Math. Soc.

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In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group $G_1$ in which all finite index subgroups are conjugacy separable, but which has an index $2$ overgroup that is not conjugacy separable. Conversely, we construct a finitely presented group $G_2$ which has a non-conjugacy separable subgroup of index $2$ such that every finite index normal overgroup of $G_2$ is conjugacy separable. The normality of the overgroup is essential in the last example, as such a group $G_2$ will always posses an index $3$ overgroup that is not conjugacy separable. Finally, we characterize $p$-conjugacy separable subdirect products of two free groups, where $p$ is a prime. We show that fibre products provide a natural correspondence between residually finite $p$-groups and $p$-conjugacy separable subdirect products of two non-abelian free groups. As a consequence, we deduce that the open question about the existence of an infinite finitely presented residually finite $p$-group is equivalent to the question about the existence of a finitely generated $p$-conjugacy separable full subdirect product of infinite index in the direct product of two free groups.Comment: v2: 38 pages; this is the version accepted for publicatio

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Commensurating endomorphisms of acylindrically hyperbolic groups and applications

Cited by 35 publications

References 50 publications

McCool groups of toral relatively hyperbolic groups

McCool groups of toral relatively hyperbolic groups

Extending higher-dimensional quasi-cocycles

On conjugacy separability of fibre products

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