Finite group actions on compact negatively curved manifolds (Q5953412)

This paper studies finite groups \(G\) acting on compact negatively curved manifolds \((M^n,g)\), where \(g\) stands for the metric. More precisely, the author investigates the diameters of the orbits \(G_x\) under the \(G\)-action. He introduces the constant \[ \eta(M^n,g,G)=\sup\{\varepsilon\mid \exists x\in M \text{ such that diam}(G_x)\geq\varepsilon\} \] and proves the following result: Let \((M^n,g)\) be a compact connected \(n\)-dimensional Riemannian manifold with sectional curvature \(K_M\) satisfying \(-1\leq K_M<0\). If \(G\) is a finite group acting effectively on \(M^n\) then \(\eta(M^n,g,G)\) has a lower bound \(\eta_n\) depending only on \(n\). Moreover, \(\eta_n>4^{-(n+4)}\).