LOWER BOUNDS FOR RESONANCES OF INFINITE-AREA RIEMANN SURFACES

For infinite-area, geometrically finite surfaces X = 0\H2, we prove new omega lower bounds on the local density of resonances D(z) when z lies in a logarithmic neighborhood of the real axis. These lower bounds involve the dimension δ of the limit set of 0. The first bound is valid when δ> 12 and shows logarithmic growth of the number D(z) of resonances at high energy, that is, when |Re(z) | → +∞. The second bound holds for δ> 34 and if 0 is an infinite-index subgroup of certain arithmetic groups. In this case we obtain a polynomial lower bound. Both results are in favor of a conjecture of Guillopé and Zworski on the existence of a fractal Weyl law for resonances. 1. Introduction an