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 \({ \overline{X} = X \cup \partial X}\) . 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.
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Alexander Lytchak was supported in part by SFB 611 and MPI für Mathematik in Bonn. Pierre-Emmanuel Caprace was supported by F.R.S.-F.N.R.S. (Fonds National de la Recherche Scientifique, Belgium).