On the virtual and residual properties of a generalization of Bestvina-Brady groups

TL;DR

This paper explores the properties of a broad class of groups generalizing Bestvina-Brady groups, providing conjectures and partial proofs about their residual finiteness, virtual torsion-freeness, and CAT(0) cubical structures.

Contribution

It introduces a new family of groups $G^M_L(S)$, formulates conjectures on their residual and virtual properties, and proves several cases, advancing understanding of their geometric and algebraic structure.

Findings

Conjectures on residual finiteness and virtual torsion-freeness of $G^M_L(S)$.

Identification of conditions under which $G^M_L(S)$ is a CAT(0) cubical group.

Partial proofs supporting the conjectures in specific cases.

Abstract

Previously one of us introduced a family of groups $G^M_L(S)$, parametrized by a finite flag complex $L$, a regular covering $M$ of $L$, and a set $S$ of integers. We give conjectural descriptions of when $G^M_L(S)$ is either residually finite or virtually torsion-free. In the case that $M$ is a finite cover and $S$ is periodic, there is an extension with kernel $G_L^M(S)$ and infinite cyclic quotient that is a CAT(0) cubical group. We conjecture that this group is virtually special. We relate these three conjectures to each other and prove many cases of them.