Spectral and dynamical invariants of Hecke triangle groups via transfer operators

In this paper, we consider Hecke triangle groups Γ w \Gamma _w for w > 2 w>2 and associated infinite-volume orbifolds with cusps Γ w ∖ H \Gamma _w \backslash \mathbb {H} . We prove that the Selberg zeta function Z Γ w ( s ) Z_{\Gamma _w}(s) can be approximated for s ∈ C ∖ ( 1 2 − N 0 ) s \in \mathbb {C} \setminus (\frac {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 explicitly compute the value of the corresponding Ruelle zeta functions at zero, obtain estimates on orders of trivial zeroes of the Selberg zeta function, and evaluate the Hausdorff dimensions of Hecke triangle groups with high precision.