Bogomolov multipliers of word labelled oriented graph groups
Mallika Roy
TL;DR
The paper develops a coherent, combinatorial framework to compute the homological Bogomolov multiplier $\widetilde{B_0}(G)$ for word labelled oriented graph (WLOG) groups and applies it to Bestvina--Brady and Artin groups. By combining Moravec's Hopf-type formula with Ratcliffe's free crossed module viewpoint, it proves finite generation and computability of $\widetilde{B_0}(G)$ from WLOG presentations, and derives explicit descriptions of $H_2(G)$ and $\widetilde{B_0}(G)$ for BB and Artin groups, without assuming the $K(\pi,1)$ conjecture in general. The results yield that $\widetilde{B_0}(G_\Gamma)$ is trivial for right-angled Artin groups and, in the BB setting, provide an explicit, computable basis for $H_2(H_{\Gamma})$ with $\widetilde{B_0}(H_{\Gamma})=0$. For Artin groups, the approach shows $H_2(A)$ has rank equal to the number of edges in the defining graph, with $\widetilde{B_0}(A)$ vanishing in the studied cases, and offers concrete examples that illustrate the method and its algorithmic nature.
Abstract
A group, whose presentation is explicitly derived in a certain way from a word labelled oriented graph (in short, WLOG), is called a WLOG group. In this work, we study homological version of Bogomolov multiplier (denoted by $\widetilde{B_0}$) for this family of groups. We prove how to compute the generators for the $\widetilde{B_0}(G)$ of a WLOG group $G$ from the underlying WLOG. We exhibit finitely presented Bestvina--Brady groups and Artin groups as WLOG groups. As applications, we compute both the multipliers: the homological version of Bogomolov multipliers and Schur multipliers, of these groups utilizing their respective WLOG group presentations. Our computation gives a new proof of the structure of the Schur multiplier of a finitely presented Bestvina--Brady group.