Torsion and closed geodesics on complex hyperbolic manifolds

Summary LetX be a compact complex manifold covered by complex hyperbolicn-space with the induced metric. Each stable horocycle has a cocomplex structure preserved by the geodesic flow. To a closed geodesic one can thus associate a piece of the Poincaré map with a holomorphic fixed point. The resulting Atiyah-Bott fixed point indices, together with the length and multiplicity of as a periodic orbit, determine the contribution of to certain zeta functionsR _p(z), 0pn. From the leading coefficient ofR _p atZ=0 and the Hodge numbersh ^ij (X) we calculate the Ray-Singer -torsionT _p (X). This indicates that the known connections between torsion and the dynamical features of closed orbits continue to hold in the holomorphic category.Corresponding results hold for the -torsion of a flat unitary bundle, extending certain formulas of Ray and Singer to the casen>1.Partially supported by the Sloan Foundation and the National Science Foundation