## Algebraic & Geometric Topology

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

Matthew C B Zaremsky

#### 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 $k ≤ n$. 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

Source
Algebr. Geom. Topol., Volume 19, Number 3 (2019), 1247-1264.

Dates
Revised: 16 May 2018
Accepted: 17 October 2018
First available in Project Euclid: 29 May 2019

https://projecteuclid.org/euclid.agt/1559095425

Digital Object Identifier
doi:10.2140/agt.2019.19.1247

Mathematical Reviews number (MathSciNet)
MR3954281

Zentralblatt MATH identifier
07078603

#### Citation

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. https://projecteuclid.org/euclid.agt/1559095425

#### References

• A Baudisch, Subgroups of semifree groups, Acta Math. Acad. Sci. Hungar. 38 (1981) 19–28
• J A Behrstock, T Januszkiewicz, W D Neumann, Quasi-isometric classification of some high dimensional right-angled Artin groups, Groups Geom. Dyn. 4 (2010) 681–692
• R Bieri, R Geoghegan, D H Kochloukova, The sigma invariants of Thompson's group $F$, Groups Geom. Dyn. 4 (2010) 263–273
• M Carr, Two-generator subgroups of right-angled Artin groups are quasi-isometrically embedded, PhD thesis, Brandeis University (2015) Available at \setbox0\makeatletter\@url https://search.proquest.com/docview/1687761198 {\unhbox0
• M Casals-Ruiz, Embeddability and quasi-isometric classification of partially commutative groups, Algebr. Geom. Topol. 16 (2016) 597–620
• M Casals-Ruiz, I Kazachkov, A Zakharov, On commensurability of right-angled Artin groups, I: RAAGs defined by trees of diameter $4$, preprint (2016)
• C H Cashen, G Levitt, Mapping tori of free group automorphisms, and the Bieri–Neumann–Strebel invariant of graphs of groups, J. Group Theory 19 (2016) 191–216
• M Clay, When does a right-angled Artin group split over $\mathbb{Z}$?, Internat. J. Algebra Comput. 24 (2014) 815–825
• M Clay, C J Leininger, D Margalit, Abstract commensurators of right-angled Artin groups and mapping class groups, Math. Res. Lett. 21 (2014) 461–467
• C Damiani, A journey through loop braid groups, Expo. Math. 35 (2017) 252–285
• M B Day, R D Wade, Subspace arrangements, BNS invariants, and pure symmetric outer automorphisms of right-angled Artin groups, Groups Geom. Dyn. 12 (2018) 173–206
• D Groves, M Hull, Abelian splittings of right-angled Artin groups, from “Hyperbolic geometry and geometric group theory” (K Fujiwara, S Kojima, K Ohshika, editors), Adv. Stud. Pure Math. 73, Math. Soc. Japan, Tokyo (2017) 159–165
• J Huang, Quasi-isometry classification of right-angled Artin groups, II: Several infinite out cases, preprint (2016)
• J Huang, Quasi-isometric classification of right-angled Artin groups, I: the finite out case, Geom. Topol. 21 (2017) 3467–3537
• J Huang, Commensurability of groups quasi-isometric to RAAGs, Invent. Math. 213 (2018) 1179–1247
• S-h Kim, T Koberda, Embedability between right-angled Artin groups, Geom. Topol. 17 (2013) 493–530
• S-H Kim, T Koberda, The geometry of the curve graph of a right-angled Artin group, Internat. J. Algebra Comput. 24 (2014) 121–169
• S-h Kim, T Koberda, Anti-trees and right-angled Artin subgroups of braid groups, Geom. Topol. 19 (2015) 3289–3306
• N Koban, J McCammond, J Meier, The BNS-invariant for the pure braid groups, Groups Geom. Dyn. 9 (2015) 665–682
• N Koban, A Piggott, The Bieri–Neumann–Strebel invariant of the pure symmetric automorphisms of a right-angled Artin group, Illinois J. Math. 58 (2014) 27–41
• C J Leininger, D Margalit, Two-generator subgroups of the pure braid group, Geom. Dedicata 147 (2010) 107–113
• J Meier, L VanWyk, The Bieri–Neumann–Strebel invariants for graph groups, Proc. London Math. Soc. 71 (1995) 263–280
• L A Orlandi-Korner, The Bieri–Neumann–Strebel invariant for basis-conjugating automorphisms of free groups, Proc. Amer. Math. Soc. 128 (2000) 1257–1262
• R Strebel, Notes on the Sigma invariants, preprint (2012)