Fractal Weyl bounds and Hecke triangle groups

Abstract

Let ${\Gamma }_w$ be a non-cofinite Hecke triangle group with cusp width $w>2$ and let $\varrho :{\Gamma }_w\to U(V)$ be a finite-dimensional unitary representation of ${\Gamma }_w$. In this note we announce a new fractal upper bound for the Selberg zeta function of ${\Gamma }_w$ twisted by $\varrho$. In strips parallel to the imaginary axis and bounded away from the real axis, the Selberg zeta function is bounded by $\exp \left(C_{\epsilon }|s{|}^{\delta +\epsilon }\right)$, where $\delta ={\delta }_w$ denotes the Hausdorff dimension of the limit set of ${\Gamma }_w$. This bound implies fractal Weyl bounds on the resonances of the Laplacian for all geometrically finite surfaces $X=\tilde{\Gamma }\backslash \mathbb{H}$ where $\tilde{\Gamma }$ is a finite index, torsion-free subgroup of ${\Gamma }_w$.

10 pages, 1 figure