Flat braid groups, right-angled Artin groups, and commensurability

Anthony Genevois

Vol. 340 (2026), No. 1, 37–70

DOI: 10.2140/pjm.2026.340.37

Abstract

For every $n\ge 1$, the flat braid group ${\mathrm{FB}<!--FUNCTION\; APPLICATION-->}_n$ is an analogue of the braid group $B_n$ that can be described as the fundamental group of the configuration space $\left\{\{x_1,\dots <!--FUNCTION\; APPLICATION-->,x_n\}\in {ℝ}^n∕\mathrm{Sym}<!--FUNCTION\; APPLICATION-->(n)\mid \text{ there exist at most two indices }i,j\text{ such that }{x}_i=x_j\right\}$. Alternatively, ${\mathrm{FB}<!--FUNCTION\; APPLICATION-->}_n$ can be described as the right-angled Coxeter group $C(P_{n-2}^{\mathrm{opp}<!--FUNCTION\; APPLICATION-->})$, where $P_{n-2}^{\mathrm{opp}<!--FUNCTION\; APPLICATION-->}$ denotes the opposite graph of the path $P_{n-2}$ of length $n-2$. We prove that, for every $n=7$ or $\ge 11$, ${\mathrm{PFB}<!--FUNCTION\; APPLICATION-->}_n$ is not virtually a right-angled Artin group, disproving a conjecture of Naik, Nanda, and Singh. In the opposite direction, we observe that ${\mathrm{FB}<!--FUNCTION\; APPLICATION-->}_7$ turns out to be commensurable to the right-angled Artin group $A(P_4)$.

Keywords

flat braid group, twin groups, planar braid group, right-angled Artin group

Mathematical Subject Classification

Primary: 20F65

Secondary: 20F36

Milestones

Received: 31 March 2025

Revised: 27 August 2025

Accepted: 18 September 2025

Published: 31 October 2025