CAT(0) spaces of higher rank II

TL;DR

This paper proves a rigidity theorem for CAT(0) spaces with higher rank, showing they are either symmetric spaces, Euclidean buildings, or split as a product, extending known results for Hadamard manifolds.

Contribution

It generalizes the Higher Rank Rigidity Theorem to CAT(0) spaces with geometric group actions, characterizing their structure under certain flatness conditions.

Findings

CAT(0) spaces with higher rank are rigid under specified conditions.

Such spaces are either symmetric spaces, Euclidean buildings, or split as a product.

The result extends the Higher Rank Rigidity Theorem beyond Hadamard manifolds.

Abstract

This belongs to a series of papers motivated by Ballmann's Higher Rank Rigidity Conjecture. We prove the following. Let $X$ be a CAT(0) space with a geometric group action. Suppose that every geodesic in $X$ lies in an $n$-flat, $n\geq 2$. If $X$ contains a periodic $n$-flat which does not bound a flat $(n+1)$-half-space, then $X$ is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product. This generalizes the Higher Rank Rigidity Theorem for Hadamard manifolds with geometric group actions.