Product set growth in groups and hyperbolic geometry

Delzant, Thomas; Steenbock, Markus

doi:10.1112/topo.12156

Cited by 12 publications

(19 citation statements)

References 21 publications

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“…In a very recent work Delzant and Steenbock [31] improve on Corollary 13.4 and on the above mentioned work of Arzhantseva-Lysienok and give an explicit entropy lower which improves as |S| gets larger.…”

Section: Corollary 134 (Uniform Growth Of Subgroups In a Hyperbolic Group)mentioning

confidence: 92%

On the joint spectral radius for isometries of non-positively curved spaces and uniform growth

Breuillard¹,

Fujiwara²

2021

Annales De l'Institut Fourier

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We recast the notion of joint spectral radius in the setting of groups acting by isometries on non-positively curved spaces and give geometric versions of results of Berger-Wang and Bochi valid for δ-hyperbolic spaces and for symmetric spaces of non-compact type. This method produces nice hyperbolic elements in many classical geometric settings. Applications to uniform growth are given, in particular a new proof and a generalization of a theorem of Besson-Courtois-Gallot.Résumé. -Nous généralisons la notion de rayon spectral joint dans le cadre des actions de groupes par isométries sur les espaces à courbure négative ou nulle et nous donnons des versions géométriques des résultats de Berger-Wang et de Bochi valables dans tout espace δ-hyperbolique ainsi que dans les espaces symétriques de type non-compact. Cette méthode permet de produire des éléments hyperboliques dans de nombreuses situations géométriques classiques. Nous donnons par ailleurs des applications à la croissance uniforme ainsi qu'une nouvelle preuve et une généralisation d'un théorème de Besson-Courtois-Gallot.

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Section: Corollary 134 (Uniform Growth Of Subgroups In a Hyperbolic Group)mentioning

confidence: 92%

On the joint spectral radius for isometries of non-positively curved spaces and uniform growth

Breuillard¹,

Fujiwara²

2021

Annales De l'Institut Fourier

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“…This estimate can be thought of as a quantified version of the Tits alternative in G. A similar statement holds for SL 2 (Z) [Cha08], free products, limit groups [But13] and groups acting on δ-hyperbolic spaces [DS20]. All these groups display strong features of negative curvature, inherited from a non-elementary acylindrical action on a hyperbolic space.…”

Section: Introductionmentioning

confidence: 71%

“…Remark 5.4. -The existence of a quasi-center is already known by [DS20]. The authors prove there that any point almost-minimizing the 1 -energy is a quasi-center.…”

Section: Energy and Quasi-centermentioning

confidence: 94%

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Product set growth in Burnside groups

Coulon¹,

Steenbock²

2022

Journal De L’École Polytechnique — Mathématiques

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Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0. https://creativecommons.org/licenses/by/4.0/ L'accès aux articles de la revue « Journal de l'École polytechnique -Mathématiques » (http://jep.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jep.centre-mersenne.org/legal/). Publié avec le soutien du Centre National de la Recherche Scientifique Publication membre du Centre Mersenne pour l'édition scientifique ouverte www.centre-mersenne.org

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“…Theorem 1.5. [DS20] Let G be a group that acts acylindrically on a hyperbolic space. Then there exist constants K > 0 and α > 0 such that for every finite U ⊂ G at least one of the following must hold:…”

Section: Historymentioning

confidence: 99%

Product set growth in mapping class groups

Kerr¹

2021

Preprint

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The main theorem in this paper is that for any group G that acts acylindrically on a quasi-tree, there exists α > 0 such that for any finite U ⊂ G that has large enough displacement in the quasi-tree, and is not contained in a virtually cyclic subgroup, we have. By a result of Balasubramanya [Bal17], the main theorem applies to every acylindrically hyperbolic group. In particular, we can show that if G is a right-angled Artin group, then there exist α, β > 0 such that for any finite symmetric U ⊂ G that is not contained in Z, and not contained non-trivially in a subgroup of the form H × Z, we have |U n | (α|U |) βn .

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Product set growth in groups and hyperbolic geometry

Cited by 12 publications

References 21 publications

On the joint spectral radius for isometries of non-positively curved spaces and uniform growth

On the joint spectral radius for isometries of non-positively curved spaces and uniform growth

Product set growth in Burnside groups

Product set growth in mapping class groups

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