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\colon\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_{\varepsilon} \vert s\vert^{\delta + \varepsilon} \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 any geometrically finite surface $ X = \widetilde{\Gamma}\backslash \mathbb{H}^2 $ whose fundamental group $ \widetilde{\Gamma} $ is a finite index, torsion-free subgroup of $ \Gamma_w $.