Affine Maps Between CAT(0) Spaces

TL;DR

This paper investigates affine maps between CAT(0) spaces with geometric actions, demonstrating they decompose into products of dilations and linear maps, and establishes a splitting lemma for the Tits boundary of such spaces.

Contribution

It extends known Riemannian results to CAT(0) spaces and introduces a splitting lemma for the Tits boundary in this context.

Findings

Affine maps decompose into dilations and linear maps on Euclidean factors

Splitting lemma for the Tits boundary of CAT(0) spaces with geometric actions

Extension of Riemannian splitting results to CAT(0) spaces

Abstract

We study affine maps between CAT(0) spaces with geometric actions, and show that they essentially split as products of dilations and linear maps (on the Euclidean factor). This extends known results from the Riemannian case. Furthermore, we prove a splitting lemma for the Tits boundary of a CAT(0) space with geometric action, a variant of a splitting lemma for geodesically complete CAT(1) spaces by Lytchak.