Algebraic & Geometric Topology

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

Matthew C B Zaremsky

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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.

Article information

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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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 20F36: Braid groups; Artin groups 57M07: Topological methods in group theory

right-angled Artin group braid group loop braid group BNS invariant abstract commensurability


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.

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