At infinity of finite-dimensional CAT(0) spaces

TL;DR

This paper proves that in finite-dimensional CAT(0) spaces, filtering families of convex sets intersect at infinity, enabling extensions of key fixed point and rigidity results from proper spaces to finite-dimensional ones.

Contribution

It establishes the intersection property at infinity for convex sets in finite-dimensional CAT(0) spaces, extending several known results to this broader setting.

Findings

Filtering families of convex sets have non-empty intersection at infinity.

Fixed points at infinity exist for parabolic isometries in finite-dimensional CAT(0) spaces.

Extensions of superrigidity and group action restrictions to finite-dimensional 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.