CAT(0) spaces with boundary the join of two Cantor sets

TL;DR

This paper characterizes the structure of certain CAT(0) spaces with boundary resembling a join of two Cantor sets, revealing their Tits boundary and group action properties, especially when geodesically complete.

Contribution

It establishes conditions under which CAT(0) spaces with boundary as a join of two Cantor sets have a specific Tits boundary and describes their geometric and group-theoretic structure.

Findings

Tits boundary is the spherical join of two uncountable discrete sets.

If geodesically complete, the space is a product of two isometry group actions.

The group contains a finite index subgroup isomorphic to a lattice in a product of trees.

Abstract

We will show that if a proper complete CAT(0) space X has a visual boundary homeomorphic to the join of two Cantor sets, and X admits a geometric group action by a group containing a subgroup isomorphic to Z^2, then its Tits boundary is the spherical join of two uncountable discrete sets. If X is geodesically complete, then X is a product, and the group has a finite index subgroup isomorphic to a lattice in the product of two isometry groups of bounded valence bushy trees.