A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries

TL;DR

This paper establishes a rank rigidity theorem for proper CAT(0) spaces with one-dimensional Tits boundaries, linking the boundary structure to the geometric and group action properties, and extending previous rigidity results.

Contribution

It proves a new rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries, including conditions under which the space is a Euclidean building or symmetric space.

Findings

If Tits diameter is π and the action is not minimal, boundary is a spherical building or join.

If the space is geodesically complete, it is a Euclidean building, symmetric space, or product.

An alternative condition involving an invariant of the group action guarantees rigidity without the diameter π assumption.

Abstract

We prove the following rank rigidity result for proper CAT(0) spaces with one-dimensional Tits boundaries: Let $\Gamma$ be a group acting properly discontinuously, cocompactly, and by isometries on such a space $X$. If the Tits diameter of $\partial X$ equals $\pi$ and $\Gamma$ does not act minimally on $\partial X$, then $\partial X$ is a spherical building or a spherical join. If $X$ is also geodesically complete, then $X$ is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of $\partial X$, does not require the Tits diameter to be $\pi$, and we give an alternate condition that guarantees rigidity when this hypothesis is removed, which is that a certain invariant of the group action be even.