At infinity of finite-dimensional CAT(0) spaces

Caprace, Pierre-Emmanuel;Lytchak, Alexander
(2010) Mathematische Annalen — Vol. 346, n° 1, p. 1-21 (2010)

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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 (X) over bar = X boolean OR partial derivative 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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Caprace, P.-E., & Lytchak, A. (2010). At infinity of finite-dimensional CAT(0) spaces. Mathematische Annalen, 346(1), 1-21. https://doi.org/10.1007/s00208-009-0381-1 (Original work published 2010)