On Subgroup Separability and Membership Problems in Twisted Right-Angled Artin Groups

TL;DR

This paper characterizes when twisted right-angled Artin groups are subgroup separable based on their defining graphs, and explores the decidability of various membership problems, extending classical results to a broader class of groups.

Contribution

It provides a graph-theoretic characterization of subgroup separability and decidability of membership problems in twisted right-angled Artin groups, generalizing previous results for classical RAAGs.

Findings

Subgroup separability characterized by absence of induced paths and squares in defining graphs.

Decidability of subgroup membership when the defining graph is chordal.

Decidability of rational and submonoid membership for certain graph classes.

Abstract

We characterize twisted right-angled Artin groups (T-RAAGs) that are subgroup separable using only their defining mixed graphs: such a group is subgroup separable if and only if the underlying simplicial graph contains neither induced paths nor squares on four vertices. This generalizes the results of Metaftsis-Raptis on classical right-angled Artin groups. Additionally, we show that the subgroup membership problem is decidable when the group is coherent, which occurs precisely when the defining mixed graph is chordal. We also address the rational and submonoid membership problems by exhibiting a cone-family of graphs for which the corresponding T-RAAGs have decidable rational and submonoid membership problems.