Fractal Weyl bounds and Hecke triangle groups

Abstract

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