Cocompact CAT(0) spaces are almost geodesically complete

Abstract

Let M be a Hadamard manifold, that is, a complete simply connected riemannian manifold with non-positive sectional curvatures. Then every geodesic segment α : [0, a] → M from α(0) to α(a) can be extended to a geodesic ray α : [0, ∞) → M. We say then that the Hadamard manifold M is geodesically complete. Note that, in this case, all geodesic rays are proper maps.CAT(0) spaces are generalizations of Hadamard manifolds. For a CAT(0) space X, all geodesic rays α : [0, ∞) → X are proper maps but, in general, X is not geodesically complete.The following definition of almost geodesic completeness was suggested by M. Mihalik:A geodesic space X, with metric d, is almost geodesically complete if there is a constant C such that for every p, q ∈ X there is a geodesic ray α : [0, ∞) → X, α(0) = p, and d(q, α) ≤ C. by isometries. If H i c (X) = 0, for some i, then X is almost geodesically complete.Theorem B. Let X be a noncompact proper CAT(0) space on which Γ acts cocompactly by isometries with discrete orbits. Then X is almost geodesically complete.Theorem B follows from theorem A and the following two propositions.Proposition A. Let X be a proper CAT(0) space on which Γ acts cocompactly by isometries with discrete orbits. Then X is properly Γ-homotopy equivalent to a Γ-finite Γ-simplicial complex K.Proposition B. Let K be a locally finite contractible simplicial complex which admits a cocompact simplicial action. Then H i c (K) = 0, for some i.Proof of Theorem B from Theorem A and Propositions A and B. Let X be a noncompact proper CAT(0) space on which Γ acts cocompactly, by isometries with discrete orbits. By Proposition A, X is properly Γ-homotopy equivalent to a Γ-finite Γ-simplicial complex K. Since every CAT(0) space is contractible, we have that K is also contractible. Hence, by Proposition A, H i c (X) = H i c (K) = 0. We can now apply Theorem A and conclude that X is almost geodesically complete.It was suggested by R. Geoghegan that proposition A above could be used to prove that that the boundary ∂X of a Γ-cocompact CAT(0) space X is a shape invariant of the Γ action. The next theorem shows that in fact this is true.Theorem C. Let X and Y be proper CAT(0) spaces on which Γ acts cocompactly by isometries with discrete orbits. If X and Y are Γ-homotopy equivalent then ∂X and ∂Y are shape equivalent.Corollary A. Let X and Y be proper CAT(0) spaces on which Γ acts cocompactly by isometries with discrete orbits. If the actions have the same isotropy, i.e. if { G < Γ : X G = ∅ } = { G < Γ : Y G = ∅ } then ∂X and ∂Y are shape equivalent.We say that a group acts on a space with finite isotropy if all isotropy groups are finite.Corollary B. Let X and Y be proper CAT(0) spaces on which Γ acts cocompactly by isometries, with discrete orbits and finite isotropy. Then ∂X and ∂Y are shape equivalent.It is known that if we assume Γ in theorem C to be hyperbolic, then in fact ∂X and ∂Y are homeomorphic (see [12]), but in general ∂X and ∂Y do not have to be homeomorphic (see [8]).Here is a short outline of the paper. In section 1 w...

Hyperbolic isometries and boundaries of systolic complexes

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