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

Abstract

We prove that if a right-angled Artin group $A_{\mathrm{\Gamma }}$ is abstractly commensurable to a group splitting nontrivially as an amalgam or HNN extension over ${\mathbb{Z}}^n$, then $A_{\mathrm{\Gamma }}$ must itself split nontrivially over ${\mathbb{Z}}^k$ for some $k\le n$. Consequently, if two right-angled Artin groups $A_{\mathrm{\Gamma }}$ and $A_{\mathrm{\Delta }}$ are commensurable and $\mathrm{\Gamma }$ has no separating $k$–cliques for any $k\le n$, then neither does $\mathrm{\Delta }$, 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 nontrivially 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.