The Bieri-Neumann-Strebel Invariants for Graph Groups

Abstract

Given a finite simplicial graph ${\cal G}$, the graph group $G{\cal is the group with generators in one-to-one correspondence with the vertices of ${\cal G}$ and with relations stating two generators commute if their associated vertices are adjacent in ${\cal G}$. The Bieri-Neumann-Strebel invariant can be explicitly described in terms of the original graph ${\cal G}$ and hence there is an explicit description of the distribution of finitely generated normal subgroups of $G{\cal G}$ with abelian quotient. We construct Eilenberg-MacLane spaces for graph groups and find partial extensions of this work to the higher dimensional invariants.