All CAT(0) Boundaries of a Group of the Form HxK are CE Equivalent

TL;DR

This paper proves that for certain CAT(0) groups splitting as a direct product with infinite factors, all their boundaries are cell-like equivalent, extending previous shape equivalence results.

Contribution

It establishes that boundaries of HxK CAT(0) groups are cell-like equivalent when the group splits as a direct product with infinite factors, using new shape theory results.

Findings

Boundaries are shape equivalent for all CAT(0) groups.

Boundaries are cell-like equivalent for HxK groups with infinite factors.

New shape theory theorem developed for this proof.

Abstract

M. Bestvina has shown that for any given torsion-free CAT(0) group G, all of its boundaries are shape equivalent. He then posed the question of whether they satisfy the stronger condition of being cell-like equivalent. In this article we prove that the answer is "Yes" in the situation where the group in question splits as a direct product with infinite factors. We accomplish this by proving an interesting theorem in shape theory.