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

Published online by Cambridge University Press:  04 August 2014

ANKE D. POHL*
Affiliation:
Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3–5, 37073 Göttingen, Germany email pohl@uni-math.gwdg.de

Abstract

By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, Möller and the present author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families:

$$\begin{eqnarray}Z(s)=\det (1-{\mathcal{L}}_{s}^{+})\det (1-{\mathcal{L}}_{s}^{-}).\end{eqnarray}$$
In this article we show that the operator families ${\mathcal{L}}_{s}^{\pm }$ arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for $s\in \mathbb{C}$, $\text{Re}s={\textstyle \frac{1}{2}}$, the operator ${\mathcal{L}}_{s}^{+}$ (respectively ${\mathcal{L}}_{s}^{-}$) has a 1-eigenfunction if and only if there exists an even (respectively odd) Maass cusp form with eigenvalue $s(1-s)$. For non-arithmetic Hecke triangle groups, this result provides a new formulation of the Phillips–Sarnak conjecture on non-existence of even Maass cusp forms.

Type
Research Article
Copyright
© Cambridge University Press, 2014 

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