Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow (Q2793103)

For hyperbolic surfaces, an approach complementary to the classical analytic and number-theoretic methods has emerged within the framework of thermodynamic formalism, in particular, transfer operator techniques. These techniques focus on the dynamics of geodesic flows rather than on the geometry of the surfaces. This has led to results which thus far have been unattained by other methods. They have also provided alternative proofs to, and different view points of, well-known results. The article in review provides instances of this.NEWLINENEWLINEHere, the author uses the idea of fast/slow systems for the billiard flow on the triangle surface underlying the Hecke triangle surfaces to show how specific weights of these systems allows one to accommodate Dirichlet, respectively von Neumann, boundary value conditions, and in turn to geometrically separate odd and even Maass cusp forms for Hecke triangle groups.NEWLINENEWLINEThe Hecke triangle groups \(\Gamma_{q} \subseteq \text{PSL}_{2}(\mathbb{R})\), \(q \in \mathbb{N}_{\geq 3}\), are Fuchsian group of the first kind and form a mixed family of arithmetic (\(q \in \{ 3, 4, 6 \}\)) and non-arithmetic (\(q \not\in \{ 3, 4, 6 \}\)) Fuchsian lattices. For \(q \in \mathbb{N}_{\geq 3}\), \(\Gamma_{q}\) is generated by the two elements NEWLINE\[NEWLINE\begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} 1 & \lambda_{q}\\ 0 & 1 \end{bmatrix},NEWLINE\]NEWLINE where \(\lambda_{q} = 2 \cos(\pi/q)\). The orbifold \(\Gamma_{q}/\mathbb{H}\) is non-compact, has one cusp at infinity and two elliptic points.NEWLINENEWLINEA Maass cusp form of \(\Gamma_{q}\) is a \(C^{\infty}\)-function \(u : \mathbb{H} \to \mathbb{C}\) which is an eigenfunction for the hyperbolic Laplace-Beltrami operator, is constant on \(\Gamma_{q}\)-orbits and, for Lebesgue almost all \(y > 0\), satisfies NEWLINE\[NEWLINE\int_{0}^{\lambda_{q}} u(x + i y) \, \mathrm{d}x = 0.NEWLINE\]NEWLINENEWLINENEWLINEThe author continues the work of her and \textit{M. Möller} [Ergodic Theory Dyn. Syst. 33, No. 1, 247--283 (2013; Zbl 1277.37048)] where a transfer operator approaches to Maass cusp forms and the Selberg zeta function for Hecke triangle groups was developed. Here, two transfer operators \(\mathcal{L}_{F, s}\) and \(\mathcal{L}_{G, s}\), respectively corresponding to the finite and accelerated discretisation of the associated geodesic flow on \(\Gamma_{q}/\mathbb{H}\), were introduced and studied. These transfer operators are, in a certain sense, compatible with the involution \(z \mapsto z^{-1}\) leading to even and odd forms \(\mathcal{L}_{F, s}^{\pm}\) and \(\mathcal{L}_{G, s}^{\pm}\) of \(\mathcal{L}_{F, s}\) and \(\mathcal{L}_{G, s}\) respectively.NEWLINENEWLINEOne aspect of the results of [loc. cit.] is that they provide a characterisation of Maass cusp forms purely in terms of classical dynamical entities. Another, is that they give rise to a formulation of the Philips-Sarnak conjecture: The space of \(1\)-eigenfunctions of \(\mathcal{L}_{F, s}^{+}\) and \(\mathcal{L}_{G, s}^{+}\), respectively \(\mathcal{L}_{F, s}^{-}\) and \(\mathcal{L}_{G, s}^{-}\), are isomorphic.NEWLINENEWLINEIn the article in review, the author's main result is a spectral version of this conjecture for all Hecke triangle groups: {For \(s \in \mathbb{C}\) with \(\text{Re}(s) > 0\), the Fredholm determinant \(\det(1 - \mathcal{L}_{G, s}^{-})\) has a zero if and only if \(s\) is the spectral parameter of an odd Maass cusp form. The Fredholm determinant \(\text{det}(1 - \mathcal{L}_{G, s}^{+})\) has a zero if and only if \(s\) is a parameter of the even spectrum.} Additionally, interesting applications are given, for example, a new formulation of the Philips-Sarnak conjecture for non-arithmetic Hecke triangle groups.NEWLINENEWLINEFor recent developments in this direction see, for instance, [the author et al., ``Isomorphisms between eigenspaces of slow and fast transfer operators'', Preprint, \url{arXiv:1606.09109}; Contemp. Math. 669, 205--236 (2016; Zbl 1376.37059)], as well as references therein.