Volume 19, issue 3 (2019)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 20
Issue 3, 1073–1600
Issue 2, 531–1072
Issue 1, 1–529

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
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