We prove that if a right-angled Artin group is abstractly commensurable to a group splitting nontrivially as an amalgam or HNN extension over , then must itself split nontrivially over for some . Consequently, if two right-angled Artin groups and are commensurable and has no separating –cliques for any , 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 the braid group is not abstractly commensurable to any group that splits nontrivially over a “free group–free” subgroup, and the same holds for for the loop braid group . Our approach makes heavy use of the Bieri–Neumann–Strebel invariant.
"Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups." Algebr. Geom. Topol. 19 (3) 1247 - 1264, 2019. https://doi.org/10.2140/agt.2019.19.1247