Z-Structures on Product Groups

A Z-structure on a group G, defined by M. Bestvina, is a pair (\hat{X}, Z) of spaces such that \hat{X} is a compact ER, Z is a Z-set in \hat{X}, G acts properly and cocompactly on X=\hat{X}\Z, and the collection of translates of any compact set in X forms a null sequence in \hat{X}. It is natural to ask whether a given group admits a Z-structure. In this paper, we will show that if two groups each admit a Z-structure, then so do their free and direct products.