Fractal Weyl bounds and Hecke triangle groups

TL;DR

This paper establishes a new fractal upper bound for the Selberg zeta function associated with non-cofinite Hecke triangle groups, leading to fractal Weyl bounds on Laplacian resonances for certain geometrically finite surfaces.

Contribution

It introduces a novel fractal upper bound for the Selberg zeta function of Hecke triangle groups twisted by unitary representations, extending understanding of spectral bounds in this setting.

Findings

Bound on Selberg zeta function in specific strips of the complex plane.

Implication of fractal Weyl bounds on Laplacian resonances.

Applicability to geometrically finite surfaces with certain subgroups.

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 all geometrically finite surfaces $X=\widetilde{\Gamma}\backslash\mathbb{H}$ where $\widetilde{\Gamma}$ is a finite index, torsion-free subgroup of $\Gamma_w$.