Rank Rigidity for CAT(0) spaces without 3-flats

TL;DR

This paper investigates the structure of CAT(0) spaces without 3-flats under group actions, revealing a dichotomy that characterizes their geometric and algebraic properties, and establishing a strong Tits Alternative for these groups.

Contribution

It proves a rank rigidity theorem for CAT(0) spaces without 3-flats, classifying their structure and implications for group actions, extending previous rigidity results.

Findings

Spaces either contain a non-flat periodic geodesic or are specific symmetric spaces or products.

Groups acting on these spaces satisfy a strong Tits Alternative.

Provides a classification of CAT(0) spaces without 3-flats based on geometric properties.

Abstract

If a group $\Gamma$ acts geometrically on a CAT(0) space $X$ without 3-flats, then either $X$ contains a $\Gamma$-periodic geodesic which does not bound a flat half-plane, or else $X$ is a rank 2 Riemannian symmetric space, a 2-dimensional Euclidean building or non-trivially splits as a metric product. Consequently all such groups satisfy a strong form of the Tits Alternative.