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On strongly quasiconvex subgroups

Hung Cong Tran

Geometry & Topology 23 (2019) 1173–1235
Abstract

We develop a theory of strongly quasiconvex subgroups of an arbitrary finitely generated group. Strong quasiconvexity generalizes quasiconvexity in hyperbolic groups and is preserved under quasi-isometry. We show that strongly quasiconvex subgroups are also more reflective of the ambient group’s geometry than the stable subgroups defined by Durham and Taylor, while still having many properties analogous to those of quasiconvex subgroups of hyperbolic groups. We characterize strongly quasiconvex subgroups in terms of the lower relative divergence of ambient groups with respect to them.

We also study strong quasiconvexity and stability in relatively hyperbolic groups, right-angled Coxeter groups, and right-angled Artin groups. We give complete descriptions of strong quasiconvexity and stability in relatively hyperbolic groups and we characterize strongly quasiconvex special subgroups and stable special subgroups of two-dimensional right-angled Coxeter groups. In the case of right-angled Artin groups, we prove that the two notions of strong quasiconvexity and stability are equivalent when the right-angled Artin group is one-ended and the subgroups have infinite index. We also characterize nontrivial strongly quasiconvex subgroups of infinite index (ie nontrivial stable subgroups) in right-angled Artin groups by quadratic lower relative divergence, expanding the work of Koberda, Mangahas, and Taylor on characterizing purely loxodromic subgroups of right-angled Artin groups.

Keywords
strongly quasiconvex subgroups, stable subgroups, strong quasiconvexity, stability
Mathematical Subject Classification 2010
Primary: 20F67, 20F65
References
Publication
Received: 21 July 2017
Revised: 13 July 2018
Accepted: 30 September 2018
Published: 28 May 2019
Proposed: Jean-Pierre Otal
Seconded: Walter Neumann, John Lott
Authors
Hung Cong Tran
Department of Mathematics
The University of Georgia
Athens, GA
United States