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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

We prove that if a right-angled Artin group AΓ is abstractly commensurable to a group splitting nontrivially as an amalgam or HNN extension over n, then AΓ must itself split nontrivially over k for some k n. Consequently, if two right-angled Artin groups AΓ and AΔ are commensurable and Γ has no separating k–cliques for any k n, then neither does Δ, 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 Bn 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 LBn. Our approach makes heavy use of the Bieri–Neumann–Strebel invariant.

right-angled Artin group, braid group, loop braid group, BNS invariant, abstract commensurability
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20F36, 57M07
Received: 28 August 2017
Revised: 16 May 2018
Accepted: 17 October 2018
Published: 21 May 2019
Matthew C B Zaremsky
Department of Mathematics and Statistics
University at Albany (SUNY)
Albany, NY
United States