Abstract
An analogue of the well-known \( \frac{3}{{16}} \) lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup L of SL(2,Z). The proof in the case that the Hausdorff of the limit set of L is bigger than \( \frac{1}{2} \) is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than \( \frac{1}{2} \) we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These “spectral gaps” are then applied to sieving problems on orbits of such groups.
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References
Borthwick, D., Spectral Theory of Infinite-Area Hyperbolic Surfaces. Progress in Mathematics, 256. Birkhäuser, Boston, MA, 2007.
Bourgain, J. & Gamburd, A., Uniform expansion bounds for Cayley graphs of SL2(Fp). Ann. of Math., 167 (2008), 625–642.
Bourgain, J., Gamburd, A. & Sarnak, P., Affine linear sieve, expanders, and sumproduct. Invent. Math., 179 (2010), 559–644.
Brooks, R., The spectral geometry of a tower of coverings. J. Differential Geom., 23 (1986), 97–107.
Burger, M., Estimation de petites valeurs propres du laplacien d’un revêtement de variétés riemanniennes compactes. C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 191–194.
Burger, M., Spectre du laplacien, graphes et topologie de Fell. Comment. Math. Helv., 63 (1988), 226–252.
Dolgopyat, D., On decay of correlations in Anosov flows. Ann. of Math., 147 (1998), 357–390.
Frobenius, G., Über Gruppencharaktere, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, pp. 985–1021. 1896.
Fuchs, E. & Sanden, K., Some experiments with integral Apollonian circle packings. Experiment. Math., 20 (2011), 380–399.
Gamburd, A., On the spectral gap for infinite index “congruence” subgroups of SL2(Z). Israel J. Math., 127 (2002), 157–200.
Guillopé, L., Lin, K. K. & Zworski, M., The Selberg zeta function for convex cocompact Schottky groups. Comm. Math. Phys., 245 (2004), 149–176.
Halberstam, H. & Richert, H.-E., Sieve Methods. London Mathematical Society Monographs, 4. Academic Press, London–New York, 1974.
Iwaniec, H. & Kowalski, E., Analytic Number Theory. American Mathematical Society Colloquium Publications, 53. Amer. Math. Soc., Providence, RI, 2004.
Kim, I., Counting, mixing and equidistribution of horospheres in geometrically finite rank one locally symmetric manifolds. Preprint, 2011. arXiv:1103.5003 [math.RT].
Kontorovich, A. & Oh, H., Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds. J. Amer. Math. Soc., 24 (2011), 603–648.
Lalley, S.P., Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits. Acta Math., 163 (1989), 1–55.
Lang, S. & Weil, A., Number of points of varieties in finite fields. Amer. J. Math., 76 (1954), 819–827.
Lax, P. D. & Phillips, R. S., The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces. J. Funct. Anal., 46 (1982), 280–350.
Magee, M., Quantitative spectral gap for thin groups of hyperbolic isometries. Preprint, 2011. arXiv:1112.2004 [math.SP].
Matthews, C. R., Vaserstein, L. N. & Weisfeiler, B., Congruence properties of Zariski-dense subgroups. I. Proc. Lond. Math. Soc., 48 (1984), 514–532.
Mazzeo, R. R. & Melrose, R. B., Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal., 75 (1987), 260–310.
Naud, F., Expanding maps on Cantor sets and analytic continuation of zeta functions. Ann. Sc. Éc. Norm. Super., 38 (2005), 116–153.
Naud, F., Precise asymptotics of the length spectrum for finite-geometry Riemann surfaces. Int. Math. Res. Not., 5 (2005), 299–310.
Parry, W. & Pollicott, M., Zeta functions and the periodic orbit structure of hyperbolic dynamics. Ast´erisque, 187–188 (1990), 268 pp.
Patterson, S. J., The limit set of a Fuchsian group. Acta Math., 136 (1976), 241–273.
Patterson, S. J., On a lattice-point problem in hyperbolic space and related questions in spectral theory. Ark. Mat., 26 (1988), 167–172.
Patterson, S. J. & Perry, P. A., The divisor of Selberg’s zeta function for Kleinian groups. Duke Math. J., 106 (2001), 321–390.
Ruelle, D., Thermodynamic Formalism. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2004.
Sansuc, J.-J., Groupe de Brauer et arithm´etique des groupes alg´ebriques lin´eaires sur un corps de nombres. J. Reine Angew. Math., 327 (1981), 12–80.
Sarnak, P., Notes on the generalized Ramanujan conjectures, in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Math. Proc., 4, pp. 659–685. Amer. Math. Soc., Providence, RI, 2005.
Sarnak, P., Integral Apollonian packings. 2009 MAA Lecture. Available at http://www.math.princeton.edu/sarnak.
Selberg, A., Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc., 20 (1956), 47–87.
Selberg, A., On the estimation of Fourier coefficients of modular forms, in Proc. Sympos. Pure Math., Vol. VIII, pp. 1–15. Amer. Math. Soc., Providence, RI, 1965.
Series, C., The infinite word problem and limit sets in Fuchsian groups. Ergodic Theory Dynam. Systems, 1 (1981), 337–360.
Sullivan, D., Discrete conformal groups and measurable dynamics. Bull. Amer. Math. Soc., 6 (1982), 57–73.
Varjú, P., Expansion in SLd(O/I), I square-free. J. Eur. Math. Soc., 14 (2012), 273–305.
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Dedicated to the memory of Atle Selberg.
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Bourgain, J., Gamburd, A. & Sarnak, P. Generalization of Selberg’s \( \frac{3}{{16}} \) theorem and affine sieve. Acta Math 207, 255–290 (2011). https://doi.org/10.1007/s11511-012-0070-x
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DOI: https://doi.org/10.1007/s11511-012-0070-x