#### Volume 11, issue 5 (2011)

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

### Carrie J Tirel

Algebraic & Geometric Topology 11 (2011) 2587–2625
##### Abstract

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

##### Keywords
$\mathcal{Z}$–structure, boundary, free product, direct product, product group
Primary: 57M07
Secondary: 20F65
##### Publication
Received: 14 October 2010
Accepted: 24 June 2011
Published: 16 September 2011
##### Authors
 Carrie J Tirel Department of Mathematics University of Wisconsin - Fox Valley 1478 Midway Rd Menasha WI 54952 USA