Lower bounds for resonances of infinite-area Riemann surfaces

For infinite-area, geometrically finite surfaces X = ‫ވ\‬ 2 , we prove new omega lower bounds on the local density of resonances Ᏸ(z) when z lies in a logarithmic neighborhood of the real axis.These lower bounds involve the dimension δ of the limit set of .The first bound is valid when δ > 1 2 and shows logarithmic growth of the number Ᏸ(z) of resonances at high energy, that is, when |Re(z)| → +∞.The second bound holds for δ > 3 4 and if 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.