$\mathcal{Z}$–Structures on product groups

Abstract

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