Proper homotopy types and Z-boundaries of spaces admitting geometric group actions

TL;DR

This paper extends geometric group theory techniques to broader settings involving proper metric ARs and torsion groups, establishing new homotopy and boundary results that unify and generalize previous findings.

Contribution

It introduces generalized theorems relating quasi-isometric groups acting on proper metric ARs, including torsion groups, and connects their boundaries via Z-structures and shape equivalence.

Findings

Proper homotopy equivalence for quasi-isometric groups on ARs.

Existence of Z-structures for groups with compactifiable boundaries.

Shape equivalence of Z-boundaries for quasi-isometric groups.

Abstract

We extend several techniques and theorems from geometric group theory so that they apply to geometric actions on arbitrary proper metric ARs (absolute retracts). A second way that we generalize earlier results is by eliminating freeness requirements often placed on the group actions. In doing so, we allow for groups with torsion. The main theorems are new in that they generalize results found in the literature, but a significant aim is expository. Toward that end, brief but reasonably comprehensive introductions to the theories of ANRs (absolute neighborhood retracts) and Z-sets are included, as well as a much shorter short introduction to shape theory. Here is a sampling of the theorems proved here. THEOREM. If quasi-isometric groups G and H act geometrically on proper metric ARs X and Y , resp., then X is proper homotopy equivalent to Y. THEOREM. If quasi-isometric groups G and H act geometrically on proper metric ARs X and Y , resp., and Y can be compactified to a Z-structure for H, then the same boundary can be added to X to obtain a Z-structure for G. THEOREM. If quasi-isometric groups G and H admit Z-structures (X^, Z_1) and (Y^, Z_2) resp., then Z_1 and Z_2 are shape equivalent.