Hecke triangle groups, transfer operators and Hausdorff dimension

We consider the family of Hecke triangle groups $Γ_{w} = \langle S, T_w\rangle$ generated by the Möbius transformations $S : z\mapsto -1/z$ and $T_{w} : z \mapsto z+w$ with $w > 2.$ In this case the corresponding hyperbolic quotient $Γ_{w}\backslash\mathbb{H}^2$ is an infinite-area orbifold. Moreover, the limit set of $Γ_w$ is a Cantor-like fractal whose Hausdorff dimension we denote by $δ(w).$ The first result of this paper asserts that the twisted Selberg zeta function $Z_{Γ_{ w}}(s, ρ)$, where $ρ: Γ_{w} \to \mathrm{U}(V)$ is an arbitrary finite-dimensional unitary representation, can be realized as the Fredholm determinant of a Mayer-type transfer operator. This result has a number of applications. We study the distribution of the zeros in the half-plane $\mathrm{Re}(s) > \frac{1}{2}$ of the Selberg zeta function of a special family of subgroups $( Γ_w^n )_{n\in \mathbb{N}}$ of $Γ_w$. These zeros correspond to the eigenvalues of the Laplacian on the associated hyperbolic surfaces $X_w^n = Γ_w^n \backslash \mathbb{H}^2$. We show that the classical Selberg zeta function $Z_{Γ_w}(s)$ can be approximated by determinants of finite matrices whose entries are explicitly given in terms of the Riemann zeta function. Moreover, we prove an asymptotic expansion for the Hausdorff dimension $δ(w)$ as $w\to \infty$.

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