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Resonances and density bounds for convex co-compact congruence subgroups of SL 2(ℤ)

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Abstract

Let Γ be a convex co-compact subgroup of SL 2(Z), and let Γ(q) be the sequence of “congruence” subgroups of Γ. Let R q ⊂ C be the resonances of the hyperbolic Laplacian on the “congruence” surfaces Γ(q)H2. We prove two results on the density of resonances in R q as q → ∞: the first shows at least C q 3 resonances in slowly growing discs, the other one is a bound from above in boxes {δ/2 < σ = Re(s) = δ}, with |Im(s) - T| = 1, where we prove a density estimate of the type \(O\left( {{T^{\delta - {\varepsilon _1}\left( \sigma \right)}}{q^{3 - {\varepsilon _2}\left( \sigma \right)}}} \right)\) with εj(σ) > 0 for all σ > δ/2, j = 1, 2. These two estimates highlight the role of the critical line \(\left\{ {\operatorname{Re} \left( s \right) = \frac{\delta }{2}} \right\}\) when looking at congruence families.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A0B9

    Dmitry Jakobson

  2. Laboratoire d’Analyse non-linéaire et Géométrie, Université d’Avignon, 33 rue Louis Pasteur, 84000, Avignon, France

    Frédéric Naud

Authors
  1. Dmitry Jakobson
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  2. Frédéric Naud
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Correspondence to Frédéric Naud.

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Jakobson, D., Naud, F. Resonances and density bounds for convex co-compact congruence subgroups of SL 2(ℤ). Isr. J. Math. 213, 443–473 (2016). https://doi.org/10.1007/s11856-016-1332-7

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  • DOI: https://doi.org/10.1007/s11856-016-1332-7

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