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

Abstract

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.

Revision, corrections and improvement of the exposition

References in corpus (10)

• The free splitting complex of a free group I: Hyperbolicity
• Induced quasi-cocycles on groups with hyperbolically embedded subgroups
• Algorithmic Properties of Relatively Hyperbolic Groups
• Small cancellation theory and Burnside problem
• Commensurating endomorphisms of acylindrically hyperbolic groups and applications
• Asymptotic dimension of relatively hyperbolic groups
• Quasi-hyperbolic planes in relatively hyperbolic groups
• Acylindrically hyperbolic groups
• Quasi-isometry rigidity of groups
• Weak hyperbolicity and free constructions

Cited by in corpus (83)

• Induced quasi-cocycles on groups with hyperbolically embedded subgroups
• Boundaries and automorphisms of hierarchically hyperbolic spaces
• Commensurating endomorphisms of acylindrically hyperbolic groups and applications
• Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups
• On strongly quasiconvex subgroups
• Growth tightness for groups with contracting elements
• Hyperbolic structures on groups
• On topologizable and non-topologizable groups
• W*-superrigidity for arbitrary actions of central quotients of braid groups
• Non simplicité du groupe de Cremona sur tout corps
• Automorphisms of graph products of groups from a geometric perspective
• A Metrizable Topology on the Contracting Boundary of a Group
• Partial periodic quotient of groups acting on a hyperbolic space
• OE and W* superrigidity results for actions by surface braid groups
• Patterson-Sullivan measures and growth of relatively hyperbolic groups
• Recognizing a relatively hyperbolic group by its Dehn fillings
• Bounded cohomology with coefficients in uniformly convex Banach spaces
• Small cancellation in acylindrically hyperbolic groups
• Product set growth in groups and hyperbolic geometry
• Quasi-hyperbolic planes in relatively hyperbolic groups
• Acylindrically hyperbolic groups with exotic properties
• Extending higher dimensional quasi-cocycles
• Deciding Isomorphy using Dehn fillings, the splitting case
• Primeness results for von Neumann algebras associated with surface braid groups
• Inner amenability for groups and central sequences in factors
• Small cancellation theory over Burnside groups
• Maximal subgroups and von Neumann subalgebras with the Haagerup property
• Cohomology of group theoretic Dehn fillings I: Cohen-Lyndon type theorems
• Acylindrical group actions on quasi-trees
• Formal conjugacy growth in acylindrically hyperbolic groups
• Boundaries of Dehn fillings
• Big mapping class groups with hyperbolic actions: classification and applications
• The normal closure of big Dehn twists, and plate spinning with rotating families
• Property $P_{naive}$ for acylindrically hyperbolic groups
• Some Applications of Group Theoretic Rips Constructions to the Classification of von Neumann Algebras
• Groups acting on CAT(0) cube complexes with uniform exponential growth
• Stable Subgroups of the Genus Two Handlebody Group
• Quasi-isometrically rigid subgroups in right-angled Coxeter groups
• Blowing up and down compacta with geometrically finite convergence actions of a group
• $C^{\ast }$-simple groups without free subgroups
• Right-angled Artin groups as normal subgroups of mapping class groups
• Random walks, WPD actions, and the Cremona group
• Transitivity degrees of countable groups and acylindrical hyperbolicity
• Maximal von Neumann subalgebras arising from maximal subgroups
• The verbal width of acylindrically hyperbolic groups is infinite
• A dynamical characterization of acylindrically hyperbolic groups
• Acylindrical hyperbolicity of the three-dimensional tame automorphism group
• Infinitely presented C(6)-groups are SQ-universal
• Controlled Floyd Separation and Non Relatively Hyperbolic Groups
• Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles
• Characteristic random subgroups of geometric groups and free abelian groups of infinite rank
• The Abelianisation of the real Cremona group
• Hyperbolically embedded virtually free subgroups of relatively hyperbolic groups
• A spectral sequence for Dehn fillings
• Aut-invariant quasimorphisms on groups
• Morse elements in Garside groups are strongly contracting
• Rank One Hilbert Geometries
• Quasi-isometry invariance of relative filling functions
• Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity
• Hyperbolically embedded subgroups and quasi-isometries of pairs
• Connected components of Morse boundaries of graphs of groups
• Random walks and quasi-convexity in acylindrically hyperbolic groups
• Parabolic subgroups acting on the additional length graph
• Orthogonal forms of Kac--Moody groups are acylindrically hyperbolic
• Height, Graded Relative Hyperbolicity and Quasiconvexity {\it (with Corrigendum)}
• Acylindrical hyperbolicity of cubical small-cancellation groups
• Rigidity of mapping class groups mod powers of twists
• Finiteness properties and Relatively Hyperbolic Groups
• On relative commutants of subalgebras in group and tracial crossed product von Neumann algebras
• Combinatorics of tight geodesics and stable lengths
• Formal solutions and the first-order theory of acylindrically hyperbolic groups
• A characterization of high transitivity for groups acting on trees
• On relative hyperbolicity for a group and relative quasiconvexity for a subgroup
• The quasi-isometry invariance of the Coset Intersection Complex
• A note on separability in outer automorphism groups
• Flat braid groups, right-angled Artin groups, and commensurability
• Notes on relatively hyperbolic groups and relatively quasiconvex subgroups
• Rigidity for von Neumann algebras of graph product groups. I. Structure of automorphisms
• Product set growth in virtual subgroups of mapping class groups
• Existence and non-existence of bounded packing in CAT(0) spaces and Gromov hyperbolic spaces
• Small cancellation and outer automorphisms of Kazhdan groups acting on hyperbolic spaces
• Invariant $C^{\ast }$-subalgebras of the reduced group $C^{\ast }$-algebra
• Vanishing of $L^2$-Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings