Erratic Boundary Behavior of CAT(0) Geodesics under G-equivariant Maps

TL;DR

This paper constructs examples of CAT(0) spaces with G-equivariant quasi-isometries where the boundary images of geodesic rays can have any prescribed finite-dimensional, connected, compact topology, demonstrating complex boundary behaviors.

Contribution

It shows that for any finite-dimensional compact space, there exist CAT(0) spaces and G-equivariant maps where geodesic boundary images realize that space's topology, revealing diverse boundary behaviors.

Findings

Boundary images can have any finite-dimensional compact topology.

Existence of G-equivariant quasi-isometries with prescribed boundary behavior.

Characterization of possible boundary image topologies.

Abstract

We show that, given any finite dimensional, connected, compact metric space Z, there exists a group G acting geometrically on two CAT(0) spaces X and Y, a G-equivariant quasi-isometry f from X to Y, and a geodesic ray c in X, such that the closure of f(c), instersected with the boundary of Y, is homeomorphic to Z. This characterizes all homeomorphism types of "geodesic boundary images" that arise in this manner.