# Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups

## TL;DR

This paper demonstrates that certain splitting properties over abelian groups are preserved under commensurability in right-angled Artin groups, and extends these invariance results to braid and loop braid groups using Bieri–Neumann–Strebel invariants.

## Contribution

It establishes commensurability invariance of splitting over abelian subgroups in right-angled Artin groups and applies similar methods to braid and loop braid groups.

## Key findings

- Splitting over $Z^k$ is a commensurability invariant in right-angled Artin groups.
- Right-angled Artin groups with no separating $k$-cliques share this property under commensurability.
- Braid and loop braid groups are not commensurable to groups splitting over free groups for certain n.

## Abstract

We prove that if a right-angled Artin group $A_\Gamma$ is abstractly commensurable to a group splitting non-trivially as an amalgam or HNN-extension over $\mathbb{Z}^n$, then $A_\Gamma$ must itself split non-trivially over $\mathbb{Z}^k$ for some $k\le n$. Consequently, if two right-angled Artin groups $A_\Gamma$ and $A_\Delta$ are commensurable and $\Gamma$ has no separating $k$-cliques for any $k\le n$ then neither does $\Delta$, so "smallest size of separating clique" is a commensurability invariant. We also discuss some implications for issues of quasi-isometry. Using similar methods we also prove that for $n\ge 4$ the braid group $B_n$ is not abstractly commensurable to any group that splits non-trivially over a "free group-free" subgroup, and the same holds for $n\ge 3$ for the loop braid group $LB_n$. Our approach makes heavy use of the Bieri--Neumann--Strebel invariant.