Fractal Weyl bounds and Hecke triangle groups

2018/10/10 by Naud, Frederic, Pohl, Anke, Soares, Louis · 1 citation
#FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1810.04489

Abstract

Let Γ_w be a non-cofinite Hecke triangle group with cusp width w>2 and let \varrho\colonΓ_w→ U(V) be a finite-dimensional unitary representation of Γ_w. In this note we announce a new fractal upper bound for the Selberg zeta function of Γ_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( C_ε \vert s\vert^{δ+ ε} ), where δ= δ_w denotes the Hausdorff dimension of the limit set of Γ_w. This bound implies fractal Weyl bounds on the resonances of the Laplacian for all geometrically finite surfaces X=\widetildeΓ\backslashℍ where \widetildeΓ is a finite index, torsion-free subgroup of Γ_w.

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• Spectral and dynamical invariants of Hecke triangle groups via transfer operators

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