Borel actions in nonpositively curved geometry and the Nielsen realisation problem

TL;DR

This paper proves a theorem linking fixed points of isometric group actions on nonpositively curved manifolds to the Nielsen realisation problem, providing a unified approach to various formulations of the problem.

Contribution

It establishes a connection between genuine and homotopy fixed points in nonpositively curved settings and applies this to unify different versions of the Nielsen realisation problem.

Findings

Genuine and homotopy fixed points coincide for certain isometric actions.

Provides a unified framework for various formulations of the Nielsen realisation problem.

Extends fixed point results to quotients of universal spaces for families.

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.