Relative hyperbolicity and relative quasiconvexity for countable groups

Abstract

We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely generated). We prove the equivalence of Gromov, Osin, and Bowditch's definitions of relative hyperbolicity for countable groups. We then give several equivalent definitions of relatively quasiconvex subgroups in terms of various natural geometries on a relatively hyperbolic group. We show that each relatively quasiconvex subgroup is itself relatively hyperbolic, and that the intersection of two relatively quasiconvex subgroups is again relatively quasiconvex. In the finitely generated case, we prove that every undistorted subgroup is relatively quasiconvex, and we compute the distortion of a finitely generated relatively quasiconvex subgroup.

44 pages, 2 figures. Version 3 addresses some minor concerns of the referee regarding the exposition. To appear in AGT

References in corpus (5)

• Algorithmic Properties of Relatively Hyperbolic Groups
• Residual finiteness, QCERF, and fillings of hyperbolic groups
• Combination of quasiconvex subgroups of relatively hyperbolic groups
• Separation of Relatively Quasiconvex Subgroups
• On Quasiconvexity and Relative Hyperbolic Structures

Cited by in corpus (103)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups
• Mixed 3-manifolds are virtually special
• Finiteness properties of cubulated groups
• On metric relative hyperbolicity
• Distinguishing geometries using finite quotients
• On strongly quasiconvex subgroups
• The Tits alternative for the automorphism group of a free product
• Separation of Relatively Quasiconvex Subgroups
• Strong accessibility for finitely presented groups
• Stallings graphs for quasi-convex subgroups
• Relations between various boundaries of relatively hyperbolic groups
• A Metrizable Topology on the Contracting Boundary of a Group
• Decidability of the Elementary Theory of a Torsion-Free Hyperbolic Group
• Partial periodic quotient of groups acting on a hyperbolic space
• Pulling back stability with applications to Out($F_n$) and relatively hyperbolic groups
• Hyperbolic groups acting improperly
• Presenting parabolic subgroups
• Patterson-Sullivan measures and growth of relatively hyperbolic groups
• Relative cubulations and groups with a 2-sphere boundary
• Recognizing a relatively hyperbolic group by its Dehn fillings
• Relativizing characterizations of Anosov subgroups, I
• Quasi-hyperbolic planes in relatively hyperbolic groups
• Virtual Amalgamation of Relatively Quasiconvex Subgroups
• Relative Quasiconvexity using Fine Hyperbolic Graphs
• Deciding Isomorphy using Dehn fillings, the splitting case
• The coarse Baum-Connes conjecture for relatively hyperbolic groups
• Floyd maps to the boundaries of relatively hyperbolic groups
• Growth of quasiconvex subgroups
• Boundaries of Dehn fillings
• Limit sets of relatively hyperbolic groups
• Quasi-isometrically rigid subgroups in right-angled Coxeter groups
• $C^{\ast }$-simple groups without free subgroups
• Free-by-cyclic groups, automorphisms and actions on nearly canonical trees
• Entropy and drift for word metric on relatively hyperbolic groups
• The large-scale geometry of right-angled Coxeter groups
• Local cut points and splittings of relatively hyperbolic groups
• Hausdorff dimension of boundaries of relatively hyperbolic groups
• A note on fine graphs and homological isoperimetric inequalities
• Relative divergence of finitely generated groups
• Hierarchies for Relatively Hyperbolic Virtually Special Groups
• Relatively geometric actions on CAT(0) cube complexes
• Distortion of surfaces in 3-manifolds
• Relative hyperbolicity of hyperbolic-by-cyclic groups
• The universal relatively hyperbolic structure on a group and relative quasiconvexity for subgroups
• The membership problem for 3-manifold groups is solvable
• Cusped spaces and quasi-isometries of relatively hyperbolic groups
• Residual properties of automorphism groups of (relatively) hyperbolic groups
• Quasi-isometric rigidity of subgroups and Filtered ends
• Statistical hyperbolicity of relatively hyperbolic groups
• Detecting geometric splittings in finitely presented groups
• Controlled Floyd Separation and Non Relatively Hyperbolic Groups
• A spectral sequence for Dehn fillings
• Geometry of the word problem for 3-manifold groups
• Ergodicity and equidistribution in Hilbert geometry
• Dismantlable classifying space for the family of parabolic subgroups of a relatively hyperbolic group
• Maps between relatively hyperbolic spaces and between their boundaries
• Boundaries of groups with isolated flats are path connected
• Bounded cohomology of classifying spaces for families of subgroups
• A topological zero-one law and elementary equivalence of finitely generated groups
• Conjugacy growth of finitely generated groups
• Algorithms detecting stability and Morseness for finitely generated groups
• Complex of Relatively Hyperbolic Groups
• Infinite periodic groups of even exponents
• Combination of fully quasiconvex subgroups and its applications
• Finiteness of Homological Filling Functions
• Farrell-Jones via Dehn fillings
• Height in Splitting of Relatively Hyperbolic Groups
• Marked length spectrum rigidity in groups with contracting elements
• Stature and separability in graphs of groups
• A small closed convex projective 4-manifold via Dehn filling
• Random walks and quasi-convexity in acylindrically hyperbolic groups
• Combination Theorems in Groups, Geometry and Dynamics
• Cohomology and the Bowditch Boundary
• Relatively hyperbolic groups with strongly shortcut parabolics are strongly shortcut
• $C^{\ast }$-simplicity for groups with non-elementary convergence group actions
• Height, Graded Relative Hyperbolicity and Quasiconvexity {\it (with Corrigendum)}
• Cubulating a free-product-by-cyclic group
• Quasi-isometric rigidity of three manifold groups
• Markov chains on hyperbolic-like groups and quasi-isometries
• Cannon--Thurston maps for CAT(0) groups with isolated flats
• On relative hyperbolicity for a group and relative quasiconvexity for a subgroup
• Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups
• Rigidity of mapping class groups mod powers of twists
• A non-quasiconvex embedding of relatively hyperbolic groups
• Quasi-Isometries, Boundaries and JSJ-Decompositions of Relatively Hyperbolic Groups
• An example of a closed 5-manifold of nonpositive curvature that fibers over a circle
• Relative cubulation of relative strict hyperbolization
• Extensions of hyperbolic groups have locally uniform exponential growth
• Hyperbolic and cubical rigidities of Thompson's group V
• Fixed subgroups of automorphisms of relatively hyperbolic groups
• Quantifying the Residual Properties of Gamma-Limit Groups
• Relative Cubulation of Small Cancellation Free Products
• Divergence spectra and Morse boundaries of relatively hyperbolic groups
• Groups Acting Acylindrically on Trees
• Prescribed virtual homological torsion of 3-manifolds
• On the quasiconvexity of the fixed subgroup of endomorphisms of relatively hyperbolic groups
• The quasi-isometry invariance of the Coset Intersection Complex
• Contracting Boundary of a Cusped Space
• Negligibility of parabolic elements in relatively hyperbolic groups
• Cocompact cubulations of mixed 3-manifolds
• Time complexity of the conjugacy problem in relatively hyperbolic groups
• Purely exponential growth of cusp-uniform actions