The Bieri-Neumann-Strebel Invariants for Graph Groups

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