  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.
