TITLE: Spectral properties of the Laplace operator and topological properties
QUESTION [17 upvotes]: Suppose that $M$ is a closed Riemannian manifold: one can construct the so called Laplace-Beltrami operator on $M$. Its spectrum contains some information of the underlying manifold: for example its dimension is encoded by the asymptotic behaviour of eigenvalues. As far as I know, there was a problem whether two isospectral closed manifolds are isometric-the negative answer was found by Milnor. So from this story I suspect that people believed that the spectrum of Laplacian will contain a lot of (geometric) information about the underlying manifold. My question is the following: 

Is it possible to extract the information about the fundamental group of $M$ from the spectrum of Laplace operator?

REPLY [3 votes]: When $M$ is negatively curved, and especially when the curvature is constant, the distribution of the eigenvalues tells something about the distribution of lengths of closed geodesics. This is because given a closed geodesic, you can construct an approximate eigenfunction. The seminal work was by A. Selberg (trace formula), improved by P. Sarnak in his PhD thesis. There have been a lot of contributors for the general theory, including Y. Colin de Verdière and more recently N. Anantharaman.