2019
DOI: 10.1016/j.jalgebra.2018.12.011
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Acylindrically hyperbolic groups with exotic properties

Abstract: We prove that every countable family of countable acylindrically hyperbolic groups has a common finitely generated acylindrically hyperbolic quotient. As an application, we obtain an acylindrically hyperbolic group Q with strong fixed point properties: Q has property F L p for all p ∈ [1, +∞), and every action of Q on a finite dimensional contractible topological space has a fixed point. In addition, Q has other properties which are rather unusual for groups exhibiting "hyperbolic-like" behaviour. E.g., Q is n… Show more

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Cited by 15 publications
(11 citation statements)
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“…Readers interested in other applications of small cancellation technique to groups with hyperbolically embedded subgroups are referred to [38] and [45]; for a slightly different approach employing rotating families see Gromov's paper [32], Coulon's survey [21], and references therein.…”
Section: 2mentioning
confidence: 99%
“…Readers interested in other applications of small cancellation technique to groups with hyperbolically embedded subgroups are referred to [38] and [45]; for a slightly different approach employing rotating families see Gromov's paper [32], Coulon's survey [21], and references therein.…”
Section: 2mentioning
confidence: 99%
“…A fruitful approach for proving algebraic, geometric, and algorithmic facts about groups is to study their actions on metric spaces which exhibit large-scale negative curvature -so-called Gromov hyperbolic metric spaces. Among many other things, such actions may be used to study quotients of groups ( [10,15]), bounded cohomology of groups ( [7]), and isoperimetric functions of their Cayley graphs.…”
Section: Introductionmentioning
confidence: 99%
“…One cannot hope to strengthen this result to encompass acylindrical actions on quasi-trees: indeed Balasubramanya in [Bal17] proved that any acylindrically hyperbolic group has an acylindrical action on a quasi-tree (see also [Osi16]), and Minasyan and Osin in [MO19] proved that some acylindrically hyperbolic groups have a fixed point property for all L p spaces.…”
Section: Introductionmentioning
confidence: 99%