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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 ℤn, then AΓ must itself split nontrivially over ℤk 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.
Keywords: right-angled Artin group, braid group, loop braid group, BNS invariant, abstract commensurability
Zaremsky, Matthew 1
@article{10_2140_agt_2019_19_1247,
author = {Zaremsky, Matthew},
title = {Commensurability invariance for abelian splittings of right-angled {Artin} groups, braid groups and loop braid groups},
journal = {Algebraic and Geometric Topology},
pages = {1247--1264},
publisher = {mathdoc},
volume = {19},
number = {3},
year = {2019},
doi = {10.2140/agt.2019.19.1247},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1247/}
}
TY - JOUR AU - Zaremsky, Matthew TI - Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups JO - Algebraic and Geometric Topology PY - 2019 SP - 1247 EP - 1264 VL - 19 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1247/ DO - 10.2140/agt.2019.19.1247 ID - 10_2140_agt_2019_19_1247 ER -
%0 Journal Article %A Zaremsky, Matthew %T Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups %J Algebraic and Geometric Topology %D 2019 %P 1247-1264 %V 19 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1247/ %R 10.2140/agt.2019.19.1247 %F 10_2140_agt_2019_19_1247
Zaremsky, Matthew. Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups. Algebraic and Geometric Topology, Tome 19 (2019) no. 3, pp. 1247-1264. doi: 10.2140/agt.2019.19.1247
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