Borel actions in nonpositively curved geometry and the Nielsen realisation problem

In this note, we record the proof of a theorem about the coincidence of genuine and homotopy fixed points for isometric group actions on complete Riemannian manifolds with nonpositive sectional curvature, and more generally, certain quotients of universal spaces for families. The result is put into context with the Nielsen realisation problem for aspherical manifolds, and we give a unifying account of different formulations of that problem, made possible by the same methods.