At infinity of finite-dimensional CAT(0) spaces

Abstract

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space $X$ has a non-empty intersection in the visual bordification $\bar{X}=X\cup \partial X$. Using this fact, several results known for proper CAT(0) spaces may be extended to finite-dimensional spaces, including the existence of canonical fixed points at infinity for parabolic isometries, algebraic and geometric restrictions on amenable group actions, and geometric superrigidity for non-elementary actions of irreducible uniform lattices in products of locally compact groups.

An erratum filling in a gap in the proof of an application of the main result has been included to the original paper

References in corpus (3)

• Isometry groups of non-positively curved spaces: structure theory
• At infinity of finite-dimensional CAT(0) spaces
• Group actions and Helly's theorem

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• Boundary maps and maximal representations on infinite dimensional Hermitian symmetric spaces
• The Polish topology of the isometry group of the infinite dimensional hyperbolic space
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