Fillings, finite generation and direct limits of relatively hyperbolic groups

Groves, Daniel; Manning, Jason Fox

doi:10.4171/ggd/16

Cited by 9 publications

(11 citation statements)

References 26 publications

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“…Applying the results from [GMS19, GM18, CF19], we prove the following relative version of Haïssinsky’s result.…”

Section: Introductionmentioning

confidence: 86%

“…The following proposition is an immediate consequence of [GM18, Corollary 6.5]. See [GM18, § 6] for the definition of -fillings, and more context.…”

Section: Relative Cubulationsmentioning

confidence: 91%

“…Proposition 2.3 provides one major benefit of relatively geometric actions over proper and cocompact (or proper and cosparse) actions. Namely, if the images of the elements of in are hyperbolic and virtually special (for example, finite or virtually cyclic) then [GM18, Theorem D] implies that is virtually special. This allows one to prove properties of by taking virtually special hyperbolic Dehn fillings and applying the properties of virtually special hyperbolic groups.…”

Section: Relative Cubulationsmentioning

confidence: 99%

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Relative cubulations and groups with a 2-sphere boundary

Einstein¹,

Groves²

2020

Compositio Math.

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“…Applying the results from [GMS19, GM18, CF19], we prove the following relative version of Haïssinsky’s result.…”

Section: Introductionmentioning

confidence: 86%

“…The following proposition is an immediate consequence of [GM18, Corollary 6.5]. See [GM18, § 6] for the definition of -fillings, and more context.…”

Section: Relative Cubulationsmentioning

confidence: 91%

Section: Relative Cubulationsmentioning

confidence: 99%

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Relative cubulations and groups with a 2-sphere boundary

Einstein¹,

Groves²

2020

Compositio Math.

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“…In the particular case when G is torsionfree, this theorem was independently proved in [18,19]. Proof.…”

Section: Group-theoretic Dehn Surgery and Normal Automorphismsmentioning

confidence: 91%

Normal automorphisms of relatively hyperbolic groups

Minasyan

Osin

2010

Trans. Amer. Math. Soc.

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Abstract. An automorphism α of a group G is normal if it fixes every normal subgroup of G setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we show that for any such group G, Inn(G) has finite index in the subgroup Aut n (G) of normal automorphisms. If, in addition, G is non-elementary and has no finite non-trivial normal subgroups, then Aut n (G) = Inn (G). As an application, we show that Out(G) is residually finite for every finitely generated residually finite group G with infinitely many ends.

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“…Definition 8.3 (compare [Groves and Manning 2007]). Let X be a connected graph with edges of length bounded below.…”

Section: The Growth Of Conjugacy Classesmentioning

confidence: 99%

Untitled

2015

Pacific J. Math.

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We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, nonelementary group G acting on a G-space X , we prove that if G contains a strongly contracting element and if G is not too badly distorted in X , then the action of G on X is a growth tight action. It follows that if X is a cocompact, relatively hyperbolic G-space, then the action of G on X is a growth tight action. This generalizes all previously known results for growth tightness of cocompact actions: every already known example of a group that admits a growth tight action and has some infinite, infinite index normal subgroups is relatively hyperbolic; conversely, relatively hyperbolic groups admit growth tight actions. This also allows us to prove that many CAT(0) groups, including flip-graph manifold groups and many right angled Artin groups, and snowflake groups admit cocompact, growth tight actions. These provide first examples of non relatively hyperbolic groups admitting interesting growth tight actions. Our main result applies as well to cusp uniform actions on hyperbolic spaces and to the action of the mapping class group on Teichmüller space with the Teichmüller metric. Towards the proof of our main result, we give equivalent characterizations of strongly contracting elements and produce new examples of group actions with strongly contracting elements. 0. Introduction 2 Part I. Growth tight actions 8 1. Preliminaries 8 2. Contraction and constriction 12 3. Abundance of strongly contracting elements 20 4. A minimal section 25 5. Embedding a free product set 26 6. The growth gap 27 7. The growth of conjugacy classes 29 MSC2010: 20E06,

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Fillings, finite generation and direct limits of relatively hyperbolic groups

Cited by 9 publications

References 26 publications

Relative cubulations and groups with a 2-sphere boundary

Relative cubulations and groups with a 2-sphere boundary

Normal automorphisms of relatively hyperbolic groups

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