Fractal Weyl bounds and Hecke triangle groups

Let ${\mathrm{\Gamma }}_w$ be a non-cofinite Hecke triangle group with cusp width $w>2$ and let $\varrho \text{\hspace{0.17em}⁣}:{\mathrm{\Gamma }}_w\to U(V)$ be a finite-dimensional unitary representation of ${\mathrm{\Gamma }}_w$. In this note we announce a new fractal upper bound for the Selberg zeta function of ${\mathrm{\Gamma }}_w$ 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 ${\mathrm{\Gamma }}_w$. This bound implies fractal Weyl bounds on the resonances of the Laplacian for all geometrically finite surfaces $X=\tilde{\mathrm{\Gamma }}\mathrm{\backslash }\mathbb{H}$ where $\tilde{\mathrm{\Gamma }}$ is a finite index, torsion-free subgroup of ${\mathrm{\Gamma }}_w$.