#### Volume 19, issue 3 (2019)

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

### Matthew C B Zaremsky

Algebraic & Geometric Topology 19 (2019) 1247–1264
##### Abstract

We prove that if a right-angled Artin group ${A}_{\Gamma }$ is abstractly commensurable to a group splitting nontrivially as an amalgam or HNN extension over ${ℤ}^{n}$, then ${A}_{\Gamma }$ must itself split nontrivially over ${ℤ}^{k}$ 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 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.

##### Keywords
right-angled Artin group, braid group, loop braid group, BNS invariant, abstract commensurability
##### Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20F36, 57M07