Acylindrically hyperbolic groups

By Denis V. Osin. 2013

371 citations recorded by the source corpus.

We say that a group $G$ is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that the class of acylindrically hyperbolic groups coincides with many other classes studied in the literature, e.g., the class $C_{geom}$ introduced by Hamenstadt, the class of groups admitting a non-elementary weakly properly discontinuous action on a hyperbolic space in the sense of Bestvina and Fujiwara, and the class of groups with hyperbolically embedded subgroups studied by Dahmani, Guirardel, and the author. We also record some basic results about acylindrically hyperbolic groups for future use.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/018c579854f66db4568e25bda8cb642376c6743a

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Boundaries and automorphisms of hierarchically hyperbolic spaces (2016, methodology): “unless diampπSpYqq ă 8. If diampπSpYqq ă 8, then Proposition 9.10 completes the proof. The HHG version requires the following theorem of Osin, which we also use elsewhere: Theorem 9.3 (Theorem 1.1 of [Osi15]). Let Gbe a group acting acylindrically on a hyperbolic space. Then exactly one of the follo”
Automorphisms of graph products of groups from a geometric perspective (2018, background): “The first property we investigate is the notion of acylindrical hyperbolicity introduced by Osin in [39], which unifies several known classes of groups with ‘negatively curved’ features such as relatively hyperbolic groups and mapping class groups (we refer to [40] for more information).”
Hyperbolicities in CAT(0) cube complexes (2017, background): “owing question: how can I prove that a given group is acylindrically hyperbolic from an action on a CAT(0) cube complex? Let us begin by recalling Osin’s deﬁnition of acylindrically hyperbolic groups [Osi16]. 13 Deﬁnition 6.1. Let G be a group acting on a metric space X. The action is acylindrical i”
Quasi-morphisms on surface diffeomorphism groups (2019, background): “ism group of a disk. In fact more is true. There is no action of Di 0(S) satisfying WPD, for if this were the case, then Di 0(S) would admit a non-elementary acylindrical action on a hyperbolic space [Osi16] and therefore have uncountably many normal subgroups [DGO17]. But Di 0(S) is known to be sim”
Acylindrical hyperbolicity of groups acting on trees (2013, positive): “Recall also that a group G is SQ-universal if every countable group embeds in a quotient of G. Theorem 3.7 (Dahmani–Guirardel–Osin [29], Osin [71]) For any acylindrically hyperbolic group G the following conditions hold.”
On strongly quasiconvex subgroups (2017, background): “In [Osi16], Osin characterizes acylindrically hyperbolic groups using hyperbolically embedded subgroups.”
Hyperbolic structures on groups (2017, background): “Recall that a group is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space [62].”
C*-simplicity and the unique trace property for discrete groups (2014, background): “The classes Creg and Dreg both properly contain the class of acylindrically hyperbolic groups introduced in [30].”
Hierarchically hyperbolic spaces I: curve complexes for cubical groups (2014, methodology): “, [BBFa]), is closely related to the Bestvina-Bromberg-Fujiwara construction [BBFb], and provides an equivalent characterization of acylindrical hyperbolicity [Osi14] which is used in a number of applications.”
Characterizations of Morse Quasi-Geodesics via Superlinear Divergence and Sublinear Contraction (2016, background): “the concept of strongly contracting projection has been a topic of intense interest in relation to mapping class groups and outer automorphisms of free groups [1, 7], acylindrically hyperbolic groups [16, 28], and contracting/Morse boundaries [30,31,13,14,25]. We introduce a more general version of ”
GROUPS ACTING ACYLINDRICALLY ON HYPERBOLIC SPACES (2017, background): “ May 2018 [33] and since then it has been one of the most active areas of research in geometric group theory. A further generalization, the class of acylindrically hyperbolic groups, was suggested in [53] and received considerable attention in the past few years. It includes many examples of interes”
Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups (2016, background): “To make this statement more precise, one possibility is to introduce acylindrically hyperbolic groups as defined in [Osi13].”
Acylindrical actions on projection complexes (2017, methodology): “ Dahmani-Guirardel-Osin start with the group acting on a δ-hyperbolic space with a WPD element and then use the projection complexes from [BBF15] to construct the hyperbolically embedded subgroup. In [Osi16], a converse is proven; a group with an inﬁnite hyperbolically embedded subgroup has an acyli”