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

We prove that if a right-angled Artin group A_Γ is abstractly commensurable to a group splitting nontrivially as an amalgam or HNN extension over Zⁿ, then A_Γ must itself split nontrivially over Z^k for some k < n. Consequently, if two right-angled Artin groups A_Γ and A_∆ are commensurable and Γ has no separating k –cliques for any k < n, then neither does ∆, 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 > 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 > 3 for the loop braid group LB_n. Our approach makes heavy use of the Bieri–Neumann–Strebel invariant.