Residual and virtual properties of generalised Bestvina-Brady groups

Abstract

This work is about classical, group-theoretic finiteness properties of a generalisation of Bestvina-Brady groups due to Ian Leary [Lea18]. Bestvina-Brady groups originally appeared sporting exotic homological finiteness properties [BB97], yet enjoy a plethora of classical properties due to naturally being subgroups of right-angled Artin groups. Closely related is the notion of special cube complexes of Haglund-Wise [HW08], which can be used to establish residual finiteness and more. Generalised Bestvina-Brady groups arise from a branching construction and are almost never special, thus the need for studying finite-index subgroups. It turns out that the property of being virtually torsion-free is key to resolving which generalised Bestvina-Brady groups are virtually special and residually finite, as it is the torsion from branched covers which causes pathological behaviour. We prove virtual specialness for different kinds of infinite families of generalised Bestvina-Brady groups [Van]. The ideas used also apply in a more general context of branching of cube complexes, and we include an application to hyperbolic groups joint with Robert Kropholler [KV]. We resolve the question of virtual torsion-freeness when the branching is governed by cyclic covers. This is done by generalising finite quotients having the structure of extensions of extraspecial groups, which were initially found by computer search. This is then further extended to any finite covers, provided the complexes involved can retract to graphs.

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Identifiers

ORCID for Ian Leary: orcid.org/0000-0001-8300-4979

ORCID for Ashot Minasyan: orcid.org/0000-0002-4986-2352

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