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Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
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vote
Hyperbolic isometries in cocompact Hadamard (i.e. cat(0) proper simply connected) spaces
There is an answer, under perhaps some conditions, in Section 6.C of
Caprace, Pierre-Emmanuel; Monod, Nicolas
Isometry groups of non-positively curved spaces: structure theory.
J. Topol. 2 (2009), no …