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2019 Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups
Matthew C B Zaremsky
Algebr. Geom. Topol. 19(3): 1247-1264 (2019). 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 n , then A Γ must itself split nontrivially over k for some k 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 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 B n 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 LB n . Our approach makes heavy use of the Bieri–Neumann–Strebel invariant.

Citation

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Matthew C B Zaremsky. "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

Information

Received: 28 August 2017; Revised: 16 May 2018; Accepted: 17 October 2018; Published: 2019
First available in Project Euclid: 29 May 2019

zbMATH: 07078603
MathSciNet: MR3954281
Digital Object Identifier: 10.2140/agt.2019.19.1247

Subjects:
Primary: 20F65
Secondary: 20F36, 57M07

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.19 • No. 3 • 2019
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