Actions with a strong fundamental domain and Sigma invariants of groups

Abstract

We show that if a group $G$ acts on a contractible $CW$-complex with a contractible strong fundamental domain and the stabilizers of the action satisfy certain properties, then the homological Sigma invariants of the group are either empty or dense in the character sphere. This was known for several families of groups like right-angled Artin groups and pure symmetric automorphisms of free groups. We also exhibit some applications including the generalization of the previous fact to pure symmetric automorphisms of right-angled Artin groups.