Relative hyperbolicity and relative quasiconvexity for countable groups

By G. Hruska, Christopher Hruska. 2008

291 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d1f1d77e1fee12f04745eeef6d64861511c4046c

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Citations in the corpus

Acylindrically hyperbolic groups (2013, background): “We explain the argument in the case when G is ﬁnitely generated and refer to [28] for the general case.”
Hyperbolicities in CAT(0) cube complexes (2017, unknown): “We refer to [Hru10] and references therein for more information on relatively hyperbolic groups.”
The virtual Haken conjecture (2012, background): “For P = P∞, a proof of the equivalence can be found in [23].”
Quasiconvexity and relatively hyperbolic groups that split (2012, methodology): “complete ﬁnite-volume manifolds of pinched negative sectional curvature. Various approaches to relative hyperbolicity were developed by Farb [6], Bowditch [3] and Osin [20], and as surveyed by Hruska [10], these notions are equivalent for ﬁnitely generated groups. We follow Bowditch’s approach: Deﬁn”
Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups. (2009, unknown): “Then by [Bo1] (see also [Hr]) the group G possesses a geometrically finite 3-discontinuous action on a compact metrizable space X.”
Dehn filling in relatively hyperbolic groups (2006, background): “Since the current paper appeared as a preprint, Chris Hruska [23] has extended the definition proposed here (Theorem 3.”
Distinguishing geometries using finite quotients (2014, background): “We refer the reader to [25] for a survey of the various equivalent definitions of relative hyperbolicity.”
On strongly quasiconvex subgroups (2017, background): “In [Hru10], Hruska proved these two definitions are equivalent to each other.”
On metric relative hyperbolicity (2012, unknown): “The following definition is taken from [Hru10].”
Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces (2011, background): “The proof of the equivalence and a detailed analysis of the case of infinitely generated groups can be found in [78].”