12/5/2003

Peter Perry
University of Kentucky

This talk concerns joint work with David Borthwick (Emory University) and Chris Judge (University of Indiana). If $X$ is a Riemann surface of finite geometry and infinite area, the Laplacian on $X$ will have at most finitely many eigenvalues, and possibly no eigenvalues, but infinitely many scattering resonances--complex numbers which like the eigenvalues correspond to normal modes of oscillation, but have nonzero imaginary part representing an exponential decay rate for the energy of the normal mode in any finite region. We define a determinant of the Laplacian whose zeros are the scattering resonances and use it to prove a compactness theorem for metrics on $X$ which have constant curvature outside a compact set and have the same eigenvalues and scattering resonances. We also prove that Selberg's zeta function for $X$ is a meromorphic function of order two for \emph{any} surface with finite geometry. As an application to the theory of discrete groups, we prove that if $\Gamma$ is a finitely generated discrete subgroup of $SL(2,\mathbb{R})$, the set of traces ${\mathrm{Tr}}(\gamma)$ determine the conjugacy class of $\Gamma$ in $SL(2,\mathbb{R})$ up to finitely many possibilities.