Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces

By Franccois Dahmani, Vincent Guirardel, Denis V. Osin. 2011

458 citations recorded by the source corpus.

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups, $Out(F_n)$, and the Cremona group. Other examples can be found among groups acting geometrically on $CAT(0)$ spaces, fundamental groups of graphs of groups, etc. We obtain a number of general results about rotating families and hyperbolically embedded subgroups; although our technique applies to a wide class of groups, it is capable of producing new results even for well-studied particular classes. For instance, we solve two open problems about mapping class groups, and obtain some results which are new even for relatively hyperbolic groups.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d6364d7223298dd8a7a496f2d2d984c36de30c0a

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Citations in the corpus

Quasi-morphisms on surface diffeomorphism groups (2019, background): “ 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 simple (since it is perfect, and [Eps70] shows that the commutat”
Growth tight actions (2014, background): “Then i→ hi.o is a Morse quasi-geodesic in X and i→ hi is a Morse quasi-geodesic in G with respect to any word metric on G. Dahmani, Guirardel, and Osin [21] define the concept of a hyperbolically embedded subgroup.”
Infinitely presented graphical small cancellation groups are acylindrically hyperbolic (2014, methodology): “Being acylindrically hyperbolic is equivalent to containing a proper infinite hyperbolically embedded subgroup in the sense of [DGO11], and the motivating examples of hyperbolically embedded subgroups are peripheral subgroups of relatively hyperbolic groups.”
Automorphisms of graph products of groups from a geometric perspective (2018, background): “One of the most striking consequences of the acylindrical hyperbolicity of a group is its SQ-universality [17], that is, every countable group embeds into a quotient of the group we are looking at.”
Small cancellation in acylindrically hyperbolic groups (2013, background): “p HG, we denote the corresponding relative metric on Hsimply by db. Recall that a metric space is called locally nite if there are nitely many elements inside any ball of nite radius. Denition 3.1. [10] Let Gbe a group, XG. We say that a collection of subgroups fH g 2 of Gis hyperbolically embedded ”
On metric relative hyperbolicity (2012, methodology): “We refer the reader to [DGO11] for the definition of hyperbolically embedded subgroup.”
Largest acylindrical actions and Stability in hierarchically hyperbolic groups (2017, methodology): “The following uses the notion of hyperbolically embedded subgroups introduced in [DGO17].”
C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-simplicity and the amenable radical (2016, unknown): “It was proved in [9] that the methods from [1] actually apply to any acylindrically hyperbolic group (therefore unifying the aforementioned results from [2,6,10,23]).”
Hyperbolicities in CAT(0) cube complexes (2017, background): “However, in [DGO17], acylindrical hyperbolicity is studied from non-necessarily cyclic subgroups called hyperbolically embedded subgroups.”
Boundaries and automorphisms of hierarchically hyperbolic spaces (2016, background): “However, G is known to be C ̊–simple under these circumstances, since G is acylindrically hyperbolic [BHS14] and has no finite normal subgroup [DGO11].”
Quasi-convexity of hyperbolically embedded subgroups (2013, methodology): “Hyperbolically embedded subgroups have been introduced by DahmaniGuirardel-Osin in [DGO11], and they give a way to capture common properties of several classes of groups including (relatively) hyperbolic groups, mapping class groups, Out(Fn), many groups acting on CAT (0) spaces and many others.”
Acylindrically hyperbolic groups (2013, methodology): “It generalizes [16, Lemma 8.11]; in our language, the latter result claims that the class of acylindrically hyperbolic groups is closed under taking normal subgroups.”
Hyperbolic structures on groups (2017, background): “ually cyclic groups acting properly on proper CAT(0) spaces and containing a rank one isometry, most 3-manifold groups, groups of deciency at least 2 and many other examples. For details we refer to [27,52,62,63] and references therein. 2.2. Induced hyperbolic structures. The proof of the second sta”
On strongly quasiconvex subgroups (2017, background): “convex subgroups and hyperbolically embedded subgroups. In particular, Dahmani-Guirardel-Osin and Sisto prove that a hyperbolically embedded subgroup is almost malnormal and strongly quasiconvex (see [DGO17] and [Sis16]) and Genevois shows that the converse holds in the case of cubulable groups [Gen”
Normal subgroups in the Cremona group (2010, methodology): “Here, we take this slightly more restrictive definition because it turns out to be sufficient for our purpose, and it makes the proof less technical; we refer to [17] or to the recent work [15] for more general viewpoints.”
Acylindrical hyperbolicity of groups acting on trees (2013, unknown): “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.”