Hecke triangle groups, transfer operators and Hausdorff dimension

TL;DR

This paper links the spectral theory of Hecke triangle groups to transfer operators, showing how Selberg zeta functions relate to determinants and providing asymptotic behavior of the Hausdorff dimension of their limit sets.

Contribution

It establishes the Fredholm determinant representation of twisted Selberg zeta functions for Hecke triangle groups and analyzes the zeros and Hausdorff dimension asymptotics.

Findings

Selberg zeta function as Fredholm determinant of transfer operator

Approximation of zeta function zeros via finite matrices and Riemann zeta function

Asymptotic expansion of Hausdorff dimension as w approaches infinity

Abstract

We consider the family of Hecke triangle groups $ \Gamma_{w} = \langle S, T_w\rangle $ generated by the M\"obius transformations $ S : z\mapsto -1/z $ and $ T_{w} : z \mapsto z+w $ with $ w > 2.$ In this case the corresponding hyperbolic quotient $ \Gamma_{w}\backslash\mathbb{H}^2 $ is an infinite-area orbifold. Moreover, the limit set of $ \Gamma_w $ is a Cantor-like fractal whose Hausdorff dimension we denote by $ \delta(w). $ The first result of this paper asserts that the twisted Selberg zeta function $ Z_{\Gamma_{ w}}(s, \rho) $, where $ \rho : \Gamma_{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 $( \Gamma_w^n )_{n\in \mathbb{N}} $ of $\Gamma_w$. These zeros correspond to the eigenvalues of the Laplacian on the associated hyperbolic surfaces $X_w^n = \Gamma_w^n \backslash \mathbb{H}^2$. We show that the classical Selberg zeta function $Z_{\Gamma_w}(s)$ 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 $\delta(w)$ as $w\to \infty$.