Finiteness properties and abelian quotients of graph groups

We describe the homological and homotopical \char'06-invariants of graph groups in terms of topological properties of sub-flag complexes of finite flag complexes. Bestvina and Brady have recently established the existence of FP groups which are not finitely presented; their examples arise as kernels of maps from graph groups to $\mathz$. Since the \char'06-invariants of a group {\it G} determine the finiteness properties of all normal subgroups above the commutator of {\it G}, our Main Theorem extends the work of Bestvina and Brady. That is, our Theorem determines the finiteness properties of kernels of maps from graph groups to abelian groups. Applications of this result are indicated.

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Published 1 January 1996