Spectral and dynamical invariants of Hecke triangle groups via transfer operators

Abstract

Abstract

In this paper, we consider Hecke triangle groups $Γ_w$ for $w>2$ and associated infinite-volume orbifolds $Γ_w \backslash \mathbb{H}$. We show that the Selberg zeta function $Z_{Γ_w}(s)$ can be approximated for $s \in \mathbb{C} \setminus \frac{1}{2}(1-2 \mathbb{N}_0)$ by determinants of finite-dimensional matrices with an explicitly computed error term that decays exponentially as the matrix size increases. As an application, we evaluate the Hausdorff dimensions of Hecke triangle groups with high precision, explicitly compute the values of the corresponding Ruelle zeta functions at zero, and obtain estimates on orders of trivial zeroes of the Selberg zeta function.