Z-Structures on Product Groups

TL;DR

This paper investigates how Z-structures, a topological concept related to group actions, are preserved under free and direct product operations of groups, expanding understanding of their structural properties.

Contribution

It proves that the free and direct products of groups admitting Z-structures also admit Z-structures, establishing new closure properties for these structures.

Findings

Z-structures are preserved under free products.

Z-structures are preserved under direct products.

Provides a method to construct Z-structures for product groups.

Abstract

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.