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

Zaremsky, Matthew

doi:10.2140/agt.2019.19.1247

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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

DOI: 10.2140/agt.2019.19.1247

Abstract

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 $ℤ^{n}$ , then $A_{Γ}$ must itself split nontrivially over $ℤ^{k}$ for some $k \leq n$ . Consequently, if two right-angled Artin groups $A_{Γ}$ and $A_{Δ}$ are commensurable and $Γ$ has no separating $k$ –cliques for any $k \leq n$ , 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 $n \geq 4$ 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 $n \geq 3$ 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

References

Publication

Received: 28 August 2017

Revised: 16 May 2018

Accepted: 17 October 2018

Published: 21 May 2019

Authors

Matthew C B Zaremsky
Department of Mathematics and Statistics University at Albany (SUNY) Albany, NY United States

Volume 19, issue 3 (2019)