Bestvina's Normal Form Complex and the Homology of Garside Groups

Abstract

A Garside group is a group admitting a finite lattice generating set $$\mathcal{D}$$ . Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π, 1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice $$\mathcal{D}$$ , and there is a simple sufficient condition that implies G is a duality group. The universal covers of these K(π, 1)s enjoy Bestvina's weak nonpositive curvature condition. Under a certain tameness condition, this implies that every solvable subgroup of G is virtually Abelian.