Fractal Weyl bounds and Hecke triangle groups

Frédéric Naud; Anke Pohl; Louis Soares

doi:10.3934/era.2019.26.003

Article Contents

2019, Volume 26: 24-35. Doi: 10.3934/era.2019.26.003

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Fractal Weyl bounds and Hecke triangle groups

1.
Laboratoire de Mathématiques D'avignon, Université d'Avignon, 301 rue Baruch de Spinoza, 84916 Avignon Cedex, France
2.
University of Bremen, Department 3 - Mathematics, Bibliothekstr. 5, 28359 Bremen, Germany
3.
Institute for Mathematics, University of Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany

Received: November 10, 2018

Revised: June 07, 2019

Published: March 2019

Naud is supported by Institut Universitaire de France. Pohl acknowledges support by the DFG grants PO 1483/2-1 and PO 1483/2-2

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Abstract

Let $ \Gamma_w $ be a non-cofinite Hecke triangle group with cusp width $ w>2 $ and let $ \varrho\colon\Gamma_w\to U(V) $ be a finite-dimensional unitary representation of $ \Gamma_w $. In this note we announce a new fractal upper bound for the Selberg zeta function of $ \Gamma_w $ twisted by $ \varrho $. In strips parallel to the imaginary axis and bounded away from the real axis, the Selberg zeta function is bounded by $ \exp\left( C_{\varepsilon} \vert s\vert^{\delta + \varepsilon} \right) $, where $ \delta = \delta_w $ denotes the Hausdorff dimension of the limit set of $ \Gamma_w. $ This bound implies fractal Weyl bounds on the resonances of the Laplacian for any geometrically finite surface $ X = \widetilde{\Gamma}\backslash \mathbb{H}^2 $ whose fundamental group $ \widetilde{\Gamma} $ is a finite index, torsion-free subgroup of $ \Gamma_w $.

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Mathematics Subject Classification: Primary: 58J50, Secondary: 37C30, 37D35, 11M36.

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Figure 1. Fundamental domain for the Hecke triangle group $ \Gamma_w $ with cusp width $ w>2 $

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