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On cubulated relatively hyperbolic groups

Eduardo Reyes

Geometry & Topology 27 (2023) 575–640
Abstract

We show that properly and cocompactly cubulated relatively hyperbolic groups are virtually special, provided the peripheral subgroups are virtually special in a way that is compatible with the cubulation. This extends Agol’s result for cubulated hyperbolic groups, and applies to a wide range of peripheral subgroups. In particular, we deduce virtual specialness for properly and cocompactly cubulated groups that are hyperbolic relative to virtually abelian groups. As another consequence, by using a theorem of Martin and Steenbock we obtain virtual specialness for groups obtained as a quotient of a free product of finitely many virtually compact special groups by a finite set of relators satisfying the classical C(1 6)–small cancellation condition.

Keywords
CAT(0) cube complexes, relatively hyperbolic groups, virtual specialness
Mathematical Subject Classification
Primary: 20F65
Secondary: 20F67, 57M07
References
Publication
Received: 2 June 2020
Revised: 1 March 2021
Accepted: 16 November 2021
Published: 16 May 2023
Proposed: Martin R Bridson
Seconded: Mladen Bestvina, David Fisher
Authors
Eduardo Reyes
Department of Mathematics
University of California at Berkeley
Berkeley, CA
United States

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