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

TL;DR

The paper analyzes flat braid groups $\mathrm{FB}_n$ and their pure versions $\mathrm{PFB}_n$ through the lens of graph products, addressing when these groups are commensurable with or virtually embed as right-angled Artin groups. It develops a toolkit based on IMC generating sets, thick subgroups, and centraliser structure to prove that $\mathrm{PFB}_n$ is not virtually a RAAG for $n=7$ or $n\ge 11$, and uses maximal-product-subgroup rigidity to extend the non-virtual-RAAG result to larger $n$. In contrast, it shows $\mathrm{FB}_7$ is commensurable to the RAAG $A(P_4)$ by relating both to lampraag models and flip manifolds, via common finite-sheeted covers. The work highlights how acylindrical hyperbolicity, centralisers in graph products, and quasi-median geometry can distinguish RAAGs from broader graph-product families, and provides methods potentially applicable to other commensurability problems between RAAGs and Coxeter-type groups.

Abstract

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

Figures (7)

• Figure 1: A flat braid from $\mathrm{FB}_3$ (namely, $\sigma_1 \sigma_2 \sigma_1$), and the two typical relations between flat braids (namely, $\sigma_1^2=1$ and $\sigma_1\sigma_3= \sigma_3 \sigma_1$.
• Figure 2: Some hyperplanes in a quasi-median graph.
• Figure 3: The flip manifold $M_1$ whose fundamental group is $A(\mathbb{L}) \rtimes \mathbb{Z}$.
• Figure 4: The flip manifold $M_2$ whose fundamental group is $A(P_4)$.
• Figure 5: The flip manifold $M_0$, a common cover of $M_1$ and $M_2$.

Theorems & Definitions (74)

• Conjecture 1.2: MR4739232
• Theorem 1.3
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• proof : Proof of Lemma \ref{['lem:WhenVirtuallyZ']}.
• Lemma 2.2
• proof
• Lemma 2.3
• Corollary 2.4