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 $A_{\Gamma }$ 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 $L{B}_n$. Our approach makes heavy use of the Bieri--Neumann--Strebel invariant.

14 pages

References in corpus (4)

• Notes on the Sigma invariants
• Quasi-isometry classification of right-angled Artin groups II: several infinite out cases
• Commensurability of groups quasi-isometric to RAAG's
• Embeddability and quasi-isometric classification of partially commutative groups

Cited by in corpus (1)

• Abelian Splittings and JSJ-Decompositions of Bestvina--Brady Groups