Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 19, Number 3 (2019), 1247-1264.
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 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.
Algebr. Geom. Topol., Volume 19, Number 3 (2019), 1247-1264.
Received: 28 August 2017
Revised: 16 May 2018
Accepted: 17 October 2018
First available in Project Euclid: 29 May 2019
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Zaremsky, Matthew C B. Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups. Algebr. Geom. Topol. 19 (2019), no. 3, 1247--1264. doi:10.2140/agt.2019.19.1247. https://projecteuclid.org/euclid.agt/1559095425