Generalizing the Croke-Kleiner Construction

TL;DR

This paper introduces a construction method for CAT(0) groups with complex, high-dimensional boundaries, aiming to explore the relationship between shape and cell-like equivalence of boundaries and address a question posed by Bestvina.

Contribution

It generalizes the Croke-Kleiner construction to produce CAT(0) groups with multiple, intricate boundaries, advancing understanding of boundary equivalences.

Findings

Constructed CAT(0) groups with multiple complex boundaries

Boundaries exhibit high-dimensional complexity

Provides new examples for boundary equivalence studies

Abstract

It is well known that every word hyperbolic group has a well-defined visual boundary. An example of C. Croke and B. Kleiner shows that the same cannot be said for CAT(0) groups. All boundaries of a CAT(0) group are, however, shape equivalent, as observed by M. Bestvina and R. Geoghegan. Bestvina has asked if they also satisfy the stronger condition of being cell-like equivalent. This article describes a construction which will produce CAT(0) groups with multiple boundaries. These groups have very complicated boundaries in high dimensions. It is our hope that their study may provide insight into Bestvina's question.