Bogomolov multipliers of word labelled oriented graph groups

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.