Large degree covers and sharp resonances of hyperbolic surfaces
Annales de l'Institut Fourier, Volume 70 (2020) no. 2, pp. 523-596.

Let Γ be a convex co-compact discrete group of isometries of the hyperbolic plane 2 , and X=Γ 2 the associated surface. In this paper we investigate the behaviour of resonances of the Laplacian Δ X ˜ for large degree covers of X given by X ˜=Γ ˜ 2 where Γ ˜Γ is a finite index normal subgroup of Γ. Using techniques of thermodynamical formalism and representation theory, we prove two new existence results of sharp non-trivial resonances close to { Re (s)=δ}, in the large degree limit, for abelian covers and infinite index congruence subgroups of SL 2 ().

On considère ici des quotients X=Γ 2 du plan hyperbolique 2 par des groupes d’isométries convexes co-compacts Γ. On s’intéresse au comportement des résonances du Laplacien Δ X ˜ X ˜=Γ ˜ 2 est un revêtement Galoisien de haut degré de X. En combinant des techniques de formalisme thermodynamique et de théorie des représentations, on prouve, dans le régime de haut degré, de nouveaux théorèmes d’existence de résonances non-triviales près de l’axe { Re (s)=δ} pour deux familles de revêtements, les cas abéliens et le cas des congruences.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3319
Classification: 58J50, 37C30, 37C35
Keywords: Hyperbolic surfaces, Geometrically finite fuchsian groups, Laplace spectrum and resonances, Selberg zeta function, Representation theory, Transfer operators and thermodynamical formalism
Jakobson, Dmitry 1; Naud, Frédéric 2; Soares, Louis 3

1 McGill University Department of Mathematics and Statistics 805 Sherbrooke Street West Montreal, Quebec, H3A0B9 (Canada)
2 Laboratoire de Mathématiques d’Avignon Avignon Université, Campus Jean-Henri Fabre, 301 rue Baruch de Spinoza 84916 Avignon Cedex 9 (France)
3 Friedrich-Schiller-Universität Jena Institut für Mathematik Ernst-Abbe-Platz 2, 07743 Jena (Germany)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2020__70_2_523_0,
     author = {Jakobson, Dmitry and Naud, Fr\'ed\'eric and Soares, Louis},
     title = {Large degree covers and sharp resonances of hyperbolic surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {523--596},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {2},
     year = {2020},
     doi = {10.5802/aif.3319},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3319/}
}
TY  - JOUR
AU  - Jakobson, Dmitry
AU  - Naud, Frédéric
AU  - Soares, Louis
TI  - Large degree covers and sharp resonances of hyperbolic surfaces
JO  - Annales de l'Institut Fourier
PY  - 2020
SP  - 523
EP  - 596
VL  - 70
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3319/
UR  - https://doi.org/10.5802/aif.3319
DO  - 10.5802/aif.3319
LA  - en
ID  - AIF_2020__70_2_523_0
ER  - 
%0 Journal Article
%A Jakobson, Dmitry
%A Naud, Frédéric
%A Soares, Louis
%T Large degree covers and sharp resonances of hyperbolic surfaces
%J Annales de l'Institut Fourier
%D 2020
%P 523-596
%V 70
%N 2
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3319
%R 10.5802/aif.3319
%G en
%F AIF_2020__70_2_523_0
Jakobson, Dmitry; Naud, Frédéric; Soares, Louis. Large degree covers and sharp resonances of hyperbolic surfaces. Annales de l'Institut Fourier, Volume 70 (2020) no. 2, pp. 523-596. doi : 10.5802/aif.3319. https://aif.centre-mersenne.org/articles/10.5802/aif.3319/

[1] Alon, Noga M.; Milman, Vitali D. λ 1 , isoperimetric inequalities for graphs, and superconcentrators, J. Comb. Theory, Ser. B, Volume 38 (1985) no. 1, pp. 73-88 | DOI | MR | Zbl

[2] Anantharaman, Nalini Precise counting results for closed orbits of Anosov flows, Ann. Sci. Éc. Norm. Supér., Volume 33 (2000) no. 1, pp. 33-56 | DOI | Numdam | MR | Zbl

[3] Babillot, Martine; Peigné, Marc Homologie des géodésiques fermées sur des variétés hyperboliques avec bouts cuspidaux, Ann. Sci. Éc. Norm. Supér., Volume 33 (2000) no. 1, pp. 81-120 | DOI | Numdam | MR | Zbl

[4] Baladi, Viviane; Vallée, Brigitte Euclidean algorithms are Gaussian, J. Number Theory, Volume 110 (2005) no. 2, pp. 331-386 | DOI | MR | Zbl

[5] Bekka, Bachir; de la Harpe, Pierre; Valette, Alain Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge University Press, 2008, xiv+472 pages | DOI | MR | Zbl

[6] Bergeron, Nicolas The spectrum of hyperbolic surfaces, Universitext, Springer; EDP Sciences, 2016, xiii+370 pages (Appendix C by Valentin Blomer and Farrell Brumley) | DOI | MR | Zbl

[7] Borthwick, David Spectral theory of infinite-area hyperbolic surfaces, Progress in Mathematics, 318, Birkhäuser, 2016, xiii+463 pages | DOI | MR | Zbl

[8] Borthwick, David; Weich, Tobias Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions, J. Spectr. Theory, Volume 6 (2016) no. 2, pp. 267-329 | DOI | MR | Zbl

[9] Bourgain, Jean; Dyatlov, Semyon Fourier dimension and spectral gaps for hyperbolic surfaces, Geom. Funct. Anal., Volume 27 (2017) no. 4, pp. 744-771 | DOI | MR | Zbl

[10] Bourgain, Jean; Gamburd, Alex; Sarnak, Peter Generalization of Selberg’s 3 16 theorem and affine sieve, Acta Math., Volume 207 (2011) no. 2, pp. 255-290 | DOI | MR | Zbl

[11] Bourgain, Jean; Kontorovich, Alex On representations of integers in thin subgroups of SL 2 (), Geom. Funct. Anal., Volume 20 (2010) no. 5, pp. 1144-1174 | DOI | MR | Zbl

[12] Bourgain, Jean; Kontorovich, Alex On Zaremba’s conjecture, Ann. Math., Volume 180 (2014) no. 1, pp. 137-196 | DOI | MR | Zbl

[13] Bowen, Lewis Free groups in lattices, Geom. Topol., Volume 13 (2009) no. 5, pp. 3021-3054 | DOI | MR | Zbl

[14] Bowen, Rufus Hausdorff dimension of quasicircles, Publ. Math., Inst. Hautes Étud. Sci. (1979) no. 50, pp. 11-25 | DOI | MR | Zbl

[15] Bowen, Rufus; Series, Caroline Markov maps associated with Fuchsian groups, Publ. Math., Inst. Hautes Étud. Sci. (1979) no. 50, pp. 153-170 | DOI | Numdam | MR | Zbl

[16] Brooks, Robert The spectral geometry of a tower of coverings, J. Differ. Geom., Volume 23 (1986) no. 1, pp. 97-107 | DOI | MR | Zbl

[17] Button, Jack All Fuchsian Schottky groups are classical Schottky groups, The Epstein birthday schrift (Geometry and Topology Monographs), Volume 1, Geometry and Topology Publications, 1998, pp. 117-125 | DOI | MR | Zbl

[18] Chang, Cheng-Hung; Mayer, Dieter H. Thermodynamic formalism and Selberg’s zeta function for modular groups, Regul. Chaotic Dyn., Volume 5 (2000) no. 3, pp. 281-312 | DOI | MR | Zbl

[19] Dal’bo, Françoise Remarques sur le spectre des longueurs d’une surface et comptages, Bol. Soc. Bras. Mat., Nova Sér., Volume 30 (1999) no. 2, pp. 199-221 | DOI | MR | Zbl

[20] Dolgopyat, Dmitry On decay of correlations in Anosov flows, Ann. Math., Volume 147 (1998) no. 2, pp. 357-390 | DOI | MR | Zbl

[21] Donnelly, Harold On the spectrum of towers, Proc. Am. Math. Soc., Volume 87 (1983) no. 2, pp. 322-329 | DOI | MR | Zbl

[22] Fedosova, Ksenia; Pohl, Anke D. Meromorphic continuation of selberg zeta functions with twists having non-expanding cusp (2017) (https://arxiv.org/abs/1709.00760) | Zbl

[23] Fell, James M. G. Weak containment and induced representations of groups, Can. J. Math., Volume 14 (1962), pp. 237-268 | DOI | MR | Zbl

[24] Folland, Gerald B. A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, 1995, x+276 pages | MR | Zbl

[25] Fried, David The zeta functions of Ruelle and Selberg. I, Ann. Sci. Éc. Norm. Supér., Volume 19 (1986) no. 4, pp. 491-517 | DOI | Numdam | MR | Zbl

[26] Gaal, Steven A. Linear analysis and representation theory, Grundlehren der Mathematischen Wissenschaften, 198, Springer, 1973, ix+688 pages | MR | Zbl

[27] Gamburd, Alex On the spectral gap for infinite index “congruence” subgroups of SL 2 (Z), Isr. J. Math., Volume 127 (2002), pp. 157-200 | DOI | MR | Zbl

[28] Gelʼfand, Israil M.; Graev, Mark I.; Pyatetskii-Shapiro, Ilya I. Representation theory and automorphic functions, Academic Press Inc., 1990, xviii+426 pages (translated from the Russian by K. A. Hirsch) | MR | Zbl

[29] de George, David L.; Wallach, Nolan R. Limit formulas for multiplicities in L 2 (ΓG), Ann. Math., Volume 107 (1978) no. 1, pp. 133-150 | DOI | MR | Zbl

[30] Guillopé, Laurent; Zworski, Maciej Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Funct. Anal., Volume 129 (1995) no. 2, pp. 364-389 | DOI | MR | Zbl

[31] Guillopé, Laurent; Zworski, Maciej Scattering asymptotics for Riemann surfaces, Ann. Math., Volume 145 (1997) no. 3, pp. 597-660 | DOI | MR | Zbl

[32] Huber, Heinz Über das Spektrum des Laplace-Operators auf kompakten Riemannschen Flächen, Comment. Math. Helv., Volume 57 (1982) no. 4, pp. 627-647 | DOI | MR | Zbl

[33] Jakobson, Dmitry; Naud, Frédéric On the critical line of convex co-compact hyperbolic surfaces, Geom. Funct. Anal., Volume 22 (2012) no. 2, pp. 352-368 | DOI | MR | Zbl

[34] Jakobson, Dmitry; Naud, Frédéric Resonances and density bounds for convex co-compact congruence subgroups of SL 2 (), Isr. J. Math., Volume 213 (2016) no. 1, pp. 443-473 | DOI | MR | Zbl

[35] Katsuda, Atsushi; Sunada, Toshikazu Homology and closed geodesics in a compact Riemann surface, Am. J. Math., Volume 110 (1988) no. 1, pp. 145-155 | DOI | MR | Zbl

[36] Katznelson, Yitzhak An introduction to harmonic analysis, Dover Publications, 1976, xiv+264 pages | MR | Zbl

[37] Lalley, Steven P. Closed geodesics in homology classes on surfaces of variable negative curvature, Duke Math. J., Volume 58 (1989) no. 3, pp. 795-821 | DOI | MR | Zbl

[38] Lalley, Steven P. Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math., Volume 163 (1989) no. 1-2, pp. 1-55 | DOI | MR | Zbl

[39] Lax, Peter D.; Phillips, Ralph S. Translation representation for automorphic solutions of the wave equation in non-Euclidean spaces. I, Commun. Pure Appl. Math., Volume 37 (1984) no. 3, pp. 303-328 | DOI | MR | Zbl

[40] Le Masson, Etienne; Sahlsten, Tuomas Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces, Duke Math. J., Volume 166 (2017) no. 18, pp. 3425-3460 | DOI | MR | Zbl

[41] Li, Jialun Fourier decay, renewal theorem and spectral gaps for random walks on split semi-simple Lie groups (2018) (https://arxiv.org/abs/1811.06484)

[42] Lubotzky, Alexander Cayley graphs: eigenvalues, expanders and random walks, Surveys in combinatorics (Stirling, 1995) (London Mathematical Society Lecture Note Series), Volume 218, Cambridge University Press, 1995, pp. 155-189 | DOI | MR | Zbl

[43] Lubotzky, Alexander Discrete groups, expanding graphs and invariant measures, Modern Birkhäuser Classics, Birkhäuser, 2010, iii+192 pages (with an appendix by Jonathan D. Rogawski) | DOI | MR | Zbl

[44] Magee, Michael Quantitative spectral gap for thin groups of hyperbolic isometries, J. Eur. Math. Soc., Volume 17 (2015) no. 1, pp. 151-187 | DOI | MR | Zbl

[45] Mayer, Dieter H. The thermodynamic formalism approach to Selberg’s zeta function for PSL (2,Z), Bull. Am. Math. Soc., Volume 25 (1991) no. 1, pp. 55-60 | DOI | MR | Zbl

[46] Mazzeo, Rafe R.; Melrose, Richard B. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., Volume 75 (1987) no. 2, pp. 260-310 | DOI | MR | Zbl

[47] McGowan, Jeffrey; Perry, Peter A. Closed geodesics in homology classes for convex co-compact hyperbolic manifolds, Proceedings of the Euroconference on Partial Differential Equations and their Applications to Geometry and Physics (Castelvecchio Pascoli, 2000), Volume 91 (2002), pp. 197-209 | DOI | MR | Zbl

[48] Naud, Frédéric Expanding maps on Cantor sets and analytic continuation of zeta functions, Ann. Sci. Éc. Norm. Supér., Volume 38 (2005) no. 1, pp. 116-153 | DOI | Numdam | MR | Zbl

[49] Naud, Frédéric Precise asymptotics of the length spectrum for finite-geometry Riemann surfaces, Int. Math. Res. Not. (2005) no. 5, pp. 299-310 | DOI | MR | Zbl

[50] Naud, Frédéric Density and location of resonances for convex co-compact hyperbolic surfaces, Invent. Math., Volume 195 (2014) no. 3, pp. 723-750 | DOI | MR | Zbl

[51] Naud, Frédéric Bornes de Weyl fractales et résonances, Séminaire Bourbaki. Vol. 2015/2016. Exposés 1104–1119 (Astérisque), Volume 390, Société Mathématique de France, 2017, pp. 77-100 | MR | Zbl

[52] Oh, Hee Eigenvalues of congruence covers of geometrically finite hyperbolic manifolds, J. Geom. Anal., Volume 25 (2015) no. 3, pp. 1421-1430 | DOI | MR | Zbl

[53] Oh, Hee; Winter, Dale Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of SL 2 (), J. Am. Math. Soc., Volume 29 (2016) no. 4, pp. 1069-1115 | DOI | MR | Zbl

[54] Parry, William; Pollicott, Mark The Chebotarov theorem for Galois coverings of Axiom A flows, Ergodic Theory Dyn. Syst., Volume 6 (1986) no. 1, pp. 133-148 | DOI | MR | Zbl

[55] Parry, William; Pollicott, Mark Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188, Société Mathématique de France, 1990 | Numdam | MR | Zbl

[56] Patterson, Samuel J. The limit set of a Fuchsian group, Acta Math., Volume 136 (1976) no. 3-4, pp. 241-273 | DOI | MR | Zbl

[57] Patterson, Samuel J.; Perry, Peter A. The divisor of Selberg’s zeta function for Kleinian groups, Duke Math. J., Volume 106 (2001) no. 2, pp. 321-390 (Appendix A by Charles Epstein) | DOI | MR | Zbl

[58] Phillips, Ralph S.; Sarnak, Peter Geodesics in homology classes, Duke Math. J., Volume 55 (1987) no. 2, pp. 287-297 | DOI | MR | Zbl

[59] Pohl, Anke D. A thermodynamic formalism approach to the Selberg zeta function for Hecke triangle surfaces of infinite area, Commun. Math. Phys., Volume 337 (2015) no. 1, pp. 103-126 | DOI | MR | Zbl

[60] Pohl, Anke D. Symbolic dynamics, automorphic functions, and Selberg zeta functions with unitary representations, Dynamics and numbers (Contemporary Mathematics), Volume 669, American Mathematical Society, 2016, pp. 205-236 | DOI | MR | Zbl

[61] Pollicott, Mark Homology and closed geodesics in a compact negatively curved surface, Am. J. Math., Volume 113 (1991) no. 3, pp. 379-385 | DOI | MR | Zbl

[62] Pollicott, Mark Some applications of thermodynamic formalism to manifolds with constant negative curvature, Adv. Math., Volume 85 (1991) no. 2, pp. 161-192 | DOI | MR | Zbl

[63] Ponomarëv, Stanislav P. Submersions and pre-images of sets of measure zero, Sib. Mat. Zh., Volume 28 (1987) no. 1, pp. 199-210 | MR | Zbl

[64] Randol, Burton Small eigenvalues of the Laplace operator on compact Riemann surfaces, Bull. Am. Math. Soc., Volume 80 (1974), pp. 996-1000 | DOI | MR | Zbl

[65] Roblin, Thomas Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr., Nouv. Sér. (2003) no. 95, vi+96 pages | DOI | Numdam | MR | Zbl

[66] Sarnak, Peter Clive Prime geodesic theorems, Ph. D. Thesis, Stanford University (USA) (1980) | MR

[67] Selberg, Atle Collected papers. I, Springer Collected Works in Mathematics, Springer, 2014, vi+711 pages (with a foreword by K. Chandrasekharan) | MR | Zbl

[68] Serre, Jean-Pierre Représentations linéaires des groupes finis, Hermann, 1978 | MR | Zbl

[69] Sharp, Richard Closed orbits in homology classes for Anosov flows, Ergodic Theory Dyn. Syst., Volume 13 (1993) no. 2, pp. 387-408 | DOI | MR | Zbl

[70] Simon, Barry Trace ideals and their applications, Mathematical Surveys and Monographs, 120, American Mathematical Society, 2005, viii+150 pages | MR | Zbl

[71] Stillwell, John Classical topology and combinatorial group theory, Graduate Texts in Mathematics, 72, Springer, 1993, xii+334 pages | DOI | MR | Zbl

[72] Suzuki, Michio Group theory. I, Grundlehren der Mathematischen Wissenschaften, 247, Springer, 1982, xiv+434 pages (translated from the Japanese by the author) | MR | Zbl

[73] Titchmarsh, Edward C. The theory of functions, Oxford University Press, 1939, x+454 pages | MR | Zbl

[74] Venkov, Alexei B.; Zograf, P. G. Analogues of Artin’s factorization formulas in the spectral theory of automorphic functions associated with induced representations of Fuchsian groups, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 46 (1982) no. 6, pp. 1150-1158 | MR | Zbl

[75] Zworski, Maciej Mathematical study of scattering resonances, Bull. Math. Sci., Volume 7 (2017) no. 1, pp. 1-85 | DOI | MR | Zbl

Cited by Sources: