CAT(0) spaces of higher rank

TL;DR

This paper proves a version of Ballmann's Rank Rigidity Conjecture for certain higher-rank CAT(0) spaces, showing they are rigid under specific geometric conditions without needing a group action.

Contribution

It confirms the conjecture for rank 2 and establishes rigidity for higher ranks under conditions involving periodic flats and geodesic containment, without requiring a geometric group action.

Findings

Confirmed the conjecture for rank 2 CAT(0) spaces.

Proved rigidity for higher ranks with specific geometric conditions.

Established rigidity without the need for a geometric group action.

Abstract

Ballmann's Rank Rigidity Conjecture predicts that a CAT(0) space of higher rank with a geometric group action is rigid -- isometric to a Riemannian symmetric space, a Euclidean building, or splits as a direct product. We confirm this conjecture in rank 2. For CAT(0) spaces of higher rank $n\geq 2$ we prove rigidity if the space contains a periodic $n$-flat and every complete geodesic lies in some $n$-flat. This does not require a geometric group action.