Borel actions in nonpositively curved geometry and the Nielsen realisation problem
Christian Kremer
TL;DR
The paper proves that a complete Riemannian manifold $M$ with nonpositive curvature and a finite group $G$ acting by isometries has genuine fixed points coincide with homotopy fixed points ($M^H \simeq M^{hH}$ for all $H \le G$), making $M$ a Borel $G$-space. It achieves this by translating the geometric problem into a homotopy-theoretic one about quotients of universal spaces for a family, showing $M \simeq \pi_1(M) \backslash E_{\mathrm{Fin}} \Gamma$ with $\Gamma=\pi_1(M_{hG})$ and $\pi=\pi_1(M)$, and proving that $ (\pi \backslash E_{\mathrm{Fin}} \Gamma)^H \simeq (\pi \backslash E_{\mathrm{Fin}} \Gamma)^{hH}$ for all $H \le G$. The central technical contribution is a purely homotopy-theoretic statement about universal spaces and their Borel quotients, established by relating orbit-category and aspherical-space models and by verifying cartesian squares that identify $\pi \backslash E_{\mathrm{Fin}} \Gamma$ with $\mathrm{Bor}(B\pi)$. The results tie into a broader Nielsen realisation program, formulating a Borel version and clarifying how manifold models for universal spaces and the equivariant Borel conjecture underpin rigidity questions for homotopical group actions.
Abstract
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.