CAT(0) spaces of higher rank I

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.