For infinite-area, geometrically finite surfaces , we prove new omega lower bounds on the local density of resonances when lies in a logarithmic neighborhood of the real axis. These lower bounds involve the dimension of the limit set of . The first bound is valid when and shows logarithmic growth of the number of resonances at high energy, that is, when . The second bound holds for 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.
"Lower bounds for resonances of infinite-area Riemann surfaces." Anal. PDE 3 (2) 207 - 225, 2010. https://doi.org/10.2140/apde.2010.3.207