Is the isometry group of a finite volume negatively curved manifold finite?
If $ M $ is compact then this is true see Why is the isometry group of a closed negatively curved manifold finite?
But what if $ M $ is finite volume noncompact? Then does negative curvature still imply finite isometry group?
Best Answer
The isometry group of a finite volume negatively curved manifold need not be finite.
Consider $B_\epsilon(0)\subset \mathbb{H}^2$ in the Poincare disk model for $0<\epsilon<1$. Since this ball is precompact, with closure contained inside of $\mathbb{H}^2$, $\mathrm{Vol}(B_{\epsilon}(0))<\infty$. However, the rotations about the origin are hyperbolic isometries and hence form a one parameter (and hence infinite) family of isometries.