On Subgroup Separability and Membership Problems in Twisted Right-Angled Artin Groups
Islam Foniqi
TL;DR
The paper characterizes subgroup separability for twisted right-angled Artin groups $T(\Gamma)$ purely from their defining mixed graphs, showing $T(\Gamma)$ is LERF if and only if the underlying graph $\overline{\Gamma}$ contains no induced $P_4$ or $C_4$ (generalizing the RAAG criterion). It then establishes that coherence (and thus decidable subgroup membership) corresponds to chordality of the mixed graph, and identifies a cone-family of mixed graphs $\mathcal{R}$ for which the rational and submonoid membership problems are decidable, leveraging a base $G=T(\Gamma\setminus\{w\})$ with a $\mathbb{Z}$-factor and closure under finite extensions. The approach combines the Reidemeister-Schreier procedure, induction on graph size, and structural decompositions (notably star-shaped subgraphs and semidirect products) to transfer the classical RAAG results to the broader T-RAAG setting. These results provide a precise link between graph-theoretic obstructions and algebraic properties like LERF and decidability, with implications for coherence and related membership problems in twisted graph groups.
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.