CAT(0) spaces of higher rank I

TL;DR

This paper proves a rigidity result for higher-rank CAT(0) spaces containing certain flats, advancing understanding of their geometric structure without requiring group actions, and focusing on Morse flats and Tits boundary properties.

Contribution

It establishes that CAT(0) spaces of rank at least 2 with a periodic flat and specific Tits boundary dimension are rigid, without needing a geometric group action.

Findings

Rigidity of CAT(0) spaces with rank ≥ 2 under certain conditions

Existence of regular points in the Tits boundary of Morse flats

Finiteness of singular points in the Tits boundary of Morse flats

Abstract

A CAT(0) space has rank at least $n$ if every geodesic lies in an $n$-flat. Ballmann's Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least $2$ with a geometric group action is rigid -- isometric to a Riemannian symmetric space, a Euclidean building, or splits as a metric product. This paper is the first in a series motivated by Ballmann's conjecture. Here we prove that a CAT(0) space of rank at least $n\geq 2$ is rigid if it contains a periodic $n$-flat and its Tits boundary has dimension $(n-1)$. This does not require a geometric group action. The result relies essentially on the study of flats which do not bound flat half-spaces -- so-called Morse flats. We show that the Tits boundary $\partial_T F$ of a periodic Morse $n$-flat $F$ contains a regular point -- a point with a Tits-neighborhood entirely contained in $\partial_T F$. More precisely, we show that the set of singular points in $\partial_T F$ can be covered by finitely many round spheres of positive codimension.