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Near optimal spectral gaps for hyperbolic surfaces. (English) Zbl 07734891

Summary: We prove that if \(X\) is a finite area non-compact hyperbolic surface, then for any \(\epsilon>0\), with probability tending to one as \(n\to\infty\), a uniformly random degree \(n\) Riemannian cover of \(X\) has no eigenvalues of the Laplacian in \([0,\frac{1}{4}-\epsilon)\) other than those of \(X\), and with the same multiplicities.
As a result, using a compactification procedure due to Buser, Burger, and Dodziuk, we settle in the affirmative the question of whether there exists a sequence of closed hyperbolic surfaces with genera tending to infinity and first non-zero eigenvalue of the Laplacian tending to \(\frac{1}{4}\).

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
30F30 Differentials on Riemann surfaces
53C20 Global Riemannian geometry, including pinching
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[1] Alon, N., Eigenvalues and expanders. Theory of Computing, Combinatorica. Combinatorica. An International Journal of the J\'{a}nos Bolyai Mathematical Society, 6, 83-96 (1986) · Zbl 0661.05053 · doi:10.1007/BF02579166
[2] Beardon, Alan F., The Geometry of Discrete Groups, Grad. Texts in Math., 91, xii+337 pp. (1983) · Zbl 0528.30001 · doi:10.1007/978-1-4612-1146-4
[3] Bergeron, Nicolas, The Spectrum of Hyperbolic Surfaces, Universitext, xiii+370 pp. (2016) · Zbl 1339.11061 · doi:10.1007/978-3-319-27666-3
[4] Bordenave, Charles; Collins, Beno\^it, Eigenvalues of random lifts and polynomials of random permutation matrices, Ann. of Math. (2). Annals of Mathematics. Second Series, 190, 811-875 (2019) · Zbl 1446.60004 · doi:10.4007/annals.2019.190.3.3
[5] Borthwick, David, Spectral Theory of Infinite-Area Hyperbolic Surfaces, Progr. Math., 318, xiii+463 pp. (2016) · Zbl 1351.58001 · doi:10.1007/978-3-319-33877-4
[6] Broder, A.; Shamir, E., On the second eigenvalue of random regular graphs. The 28th Annual Symposium on Foundations of Computer Science, 286-294 (1987) · doi:10.1109/SFCS.1987.45
[7] Brooks, Robert; Makover, Eran, Riemann surfaces with large first eigenvalue, J. Anal. Math.. Journal d’Analyse Math\'{e}matique, 83, 243-258 (2001) · Zbl 0981.30031 · doi:10.1007/BF02790263
[8] Brooks, Robert; Makover, Eran, Random construction of {R}iemann surfaces, J. Differential Geom.. Journal of Differential Geometry, 68, 121-157 (2004) · Zbl 1095.30037 · doi:10.4310/jdg/1102536712
[9] Buser, Peter, Cubic graphs and the first eigenvalue of a {R}iemann surface, Math. Z.. Mathematische Zeitschrift, 162, 87-99 (1978) · Zbl 0371.53032 · doi:10.1007/BF01437826
[10] Buser, Peter, On the bipartition of graphs, Discrete Appl. Math.. Discrete Applied Mathematics. The Journal of Combinatorial Algorithms, Informatics and Computational Sciences, 9, 105-109 (1984) · Zbl 0544.05038 · doi:10.1016/0166-218X(84)90093-3
[11] Buser, Peter, Geometry and Spectra of Compact {R}iemann Surfaces, Progr. in Math., 106, xiv+454 pp. (1992) · Zbl 0770.53001
[12] Buser, Peter; Burger, Marc; Dodziuk, Jozef, Riemann surfaces of large genus and large {\( \lambda_1\)}. Geometry and Analysis on Manifolds, Lecture Notes in Math., 1339, 54-63 (1988) · Zbl 0646.53040 · doi:10.1007/BFb0083046
[13] Collins, Beno\^it; Male, Camille, The strong asymptotic freeness of {H}aar and deterministic matrices, Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 47, 147-163 (2014) · Zbl 1303.15043 · doi:10.24033/asens.2211
[14] Dixon, John D., The probability of generating the symmetric group, Math. Z.. Mathematische Zeitschrift, 110, 199-205 (1969) · Zbl 0176.29901 · doi:10.1007/BF01110210
[15] Friedman, Joel, Relative expanders or weakly relatively {R}amanujan graphs, Duke Math. J.. Duke Mathematical Journal, 118, 19-35 (2003) · Zbl 1035.05058 · doi:10.1215/S0012-7094-03-11812-8
[16] Friedman, Joel, A proof of {A}lon’s second eigenvalue conjecture and related problems, Mem. Amer. Math. Soc.. Memoirs of the American Mathematical Society, 195, viii+100 pp. (2008) · Zbl 1177.05070 · doi:10.1090/memo/0910
[17] Gelbart, Stephen; Jacquet, Herv\'{e}, A relation between automorphic representations of {\({\rm GL}(2)\)} and {\({\rm GL}(3)\)}, Ann. Sci. \'{E}cole Norm. Sup. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 11, 471-542 (1978) · Zbl 0406.10022 · doi:10.24033/asens.1355
[18] Guillop\'{e}, Laurent; Zworski, Maciej, Upper bounds on the number of resonances for non-compact {R}iemann surfaces, J. Funct. Anal.. Journal of Functional Analysis, 129, 364-389 (1995) · Zbl 0841.58063 · doi:10.1006/jfan.1995.1055
[19] Haagerup, Uffe; Thorbj{\o}rnsen, Steen, A new application of random matrices: {\({\rm Ext}(C^*_{\rm red}(F_2))\)} is not a group, Ann. of Math. (2). Annals of Mathematics. Second Series, 162, 711-775 (2005) · Zbl 1103.46032 · doi:10.4007/annals.2005.162.711
[20] Hide, W., Spectral gap for {W}eil–{P}etersson random surfaces with cusps, Internat. Math. Res. Not., rnac293 pp. (2022) · Zbl 07794949 · doi:10.1093/imrn/rnac293
[21] Huber, Heinz, \"{U}ber den ersten {E}igenwert des {L}aplace-{O}perators auf kompakten {R}iemannschen {F}l\"{a}chen, Comment. Math. Helv.. Commentarii Mathematici Helvetici, 49, 251-259 (1974) · Zbl 0287.35075 · doi:10.1007/BF02566733
[22] Jacquet, H.; Langlands, R. P., Automorphic {F}orms on {\({\rm GL}(2)\)}, Lecture Notes in Math., 114, vii+548 pp. (1970) · Zbl 0236.12010 · doi:10.1007/BFb0058988
[23] Kim, Henry H., Functoriality for the exterior square of {\({\rm GL}_4\)} and the symmetric fourth of {\({\rm GL}_2\)}, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 16, 139-183 (2003) · Zbl 1018.11024 · doi:10.1090/S0894-0347-02-00410-1
[24] Kim, Henry H.; Shahidi, Freydoon, Cuspidality of symmetric powers with applications, Duke Math. J.. Duke Mathematical Journal, 112, 177-197 (2002) · Zbl 1074.11027 · doi:10.1215/S0012-9074-02-11215-0
[25] Kim, Henry H.; Shahidi, Freydoon, Functorial products for {\({\rm GL}_2\times{\rm GL}_3\)} and the symmetric cube for {\({\rm GL}_2\)}, Ann. of Math. (2). Annals of Mathematics. Second Series, 155, 837-893 (2002) · Zbl 1040.11036 · doi:10.2307/3062134
[26] Lax, Peter D.; Phillips, Ralph S., The asymptotic distribution of lattice points in {E}uclidean and non-{E}uclidean spaces. Functional Analysis and Approximation, Internat. Ser. Numer. Math., 60, 373-383 (1981) · Zbl 0467.10037 · doi:10.1007/978-3-0348-9369-5_34
[27] Lipnowski, A.; Wright, A., Towards optimal spectral gaps in large genus (2021)
[28] Luo, W.; Rudnick, Z.; Sarnak, P., On {S}elberg’s eigenvalue conjecture, Geom. Funct. Anal.. Geometric and Functional Analysis, 5, 387-401 (1995) · Zbl 0844.11038 · doi:10.1007/BF01895672
[29] Magee, Michael, Quantitative spectral gap for thin groups of hyperbolic isometries, J. Eur. Math. Soc. (JEMS). Journal of the European Mathematical Society (JEMS), 17, 151-187 (2015) · Zbl 1332.11058 · doi:10.4171/JEMS/500
[30] Magee, Michael; Naud, Fr\'{e}d\'{e}ric, Explicit spectral gaps for random covers of {R}iemann surfaces, Publ. Math. Inst. Hautes \'{E}tudes Sci.. Publications Math\'{e}matiques. Institut de Hautes \'{E}tudes Scientifiques, 132, 137-179 (2020) · Zbl 1508.58008 · doi:10.1007/s10240-020-00118-w
[31] Magee, Michael; Naud, Fr\'{e}d\'{e}ric, Extension of {A}lon’s and {F}riedman’s conjectures to {S}chottky surfaces (2021)
[32] Magee, Michael; Naud, Fr\'{e}d\'{e}ric; Puder, Doron, A random cover of a compact hyperbolic surface has relative spectral gap {\( \frac{3}{16}-\varepsilon \)}, Geom. Funct. Anal.. Geometric and Functional Analysis, 32, 595-661 (2022) · Zbl 1498.58022 · doi:10.1007/s00039-022-00602-x
[33] Magee, Michael; Puder, Doron, The asymptotic statistics of random covering surfaces (2020) · Zbl 1521.20016
[34] McKean, H. P., Selberg’s trace formula as applied to a compact {R}iemann surface, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 25, 225-246 (1972) · doi:10.1002/cpa.3160250302
[35] McKean, H. P., Correction to: “{S}elberg”s trace formula as applied to a compact {R}iemann surface” ({C}omm. {P}ure {A}ppl. {M}ath. {\bf 25} (1972), 225-246), Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 27, 134 pp. (1974) · Zbl 0317.30018 · doi:10.1002/cpa.3160270109
[36] Mirzakhani, Maryam, Growth of {W}eil-{P}etersson volumes and random hyperbolic surfaces of large genus, J. Differential Geom.. Journal of Differential Geometry, 94, 267-300 (2013) · Zbl 1270.30014 · doi:10.4310/jdg/1367438650
[37] Otal, Jean-Pierre; Rosas, Eulalio, Pour toute surface hyperbolique de genre {\(g,\ \lambda_{2g-2}>1/4\)}, Duke Math. J.. Duke Mathematical Journal, 150, 101-115 (2009) · Zbl 1179.30041 · doi:10.1215/00127094-2009-048
[38] Pisier, Gilles, A simple proof of a theorem of {K}irchberg and related results on {\(C^*\)}-norms, J. Operator Theory. Journal of Operator Theory, 35, 317-335 (1996) · Zbl 0858.46045
[39] Pisier, Gilles, On a linearization trick, Enseign. Math.. L’Enseignement Math\'{e}matique, 64, 315-326 (2018) · Zbl 1462.46063 · doi:10.4171/LEM/64-3/4-5
[40] Randol, Burton, Small eigenvalues of the {L}aplace operator on compact {R}iemann surfaces, Bull. Amer. Math. Soc.. Bulletin of the American Mathematical Society, 80, 996-1000 (1974) · Zbl 0317.30017 · doi:10.1090/S0002-9904-1974-13609-8
[41] Selberg, A., Harmonic analysis and discontinuous groups in weakly symmetric {R}iemannian spaces with applications to {D}irichlet series, J. Indian Math. Soc. (N.S.). The Journal of the Indian Mathematical Society. New Series, 20, 47-87 (1956) · Zbl 0072.08201
[42] Selberg, Atle, On the estimation of {F}ourier coefficients of modular forms. Proc. {S}ympos. {P}ure {M}ath., {V}ol. {VIII}, 1-15 (1965) · Zbl 0142.33903 · doi:10.1090/pspum/008/0182610
[43] Strichartz, Robert S., Analysis of the {L}aplacian on the complete {R}iemannian manifold, J. Functional Analysis. Journal of Functional Analysis, 52, 48-79 (1983) · Zbl 0515.58037 · doi:10.1016/0022-1236(83)90090-3
[44] Vodev, Georgi, Sharp polynomial bounds on the number of scattering poles for metric perturbations of the {L}aplacian in {\({\bf R}^n\)}, Math. Ann.. Mathematische Annalen, 291, 39-49 (1991) · Zbl 0754.35105 · doi:10.1007/BF01445189
[45] Vodev, Georgi, Sharp bounds on the number of scattering poles for perturbations of the {L}aplacian, Comm. Math. Phys.. Communications in Mathematical Physics, 146, 205-216 (1992) · Zbl 0766.35032 · doi:10.1007/BF02099213
[46] Vodev, Georgi, Sharp bounds on the number of scattering poles in even-dimensional spaces, Duke Math. J.. Duke Mathematical Journal, 74, 1-17 (1994) · Zbl 0813.35075 · doi:10.1215/S0012-7094-94-07401-2
[47] Weil, Andr\'{e}, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A.. Proceedings of the National Academy of Sciences of the United States of America, 34, 204-207 (1948) · Zbl 0032.26102 · doi:10.1073/pnas.34.5.204
[48] Wright, Alex, A tour through {M}irzakhani’s work on moduli spaces of {R}iemann surfaces, Bull. Amer. Math. Soc. (N.S.). American Mathematical Society. Bulletin. New Series, 57, 359-408 (2020) · Zbl 1452.32003 · doi:10.1090/bull/1687
[49] Wu, Yunhui; Xue, Yuhao, Small eigenvalues of closed {R}iemann surfaces for large genus, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 375, 3641-3663 (2022) · Zbl 1487.58031 · doi:10.1090/tran/8608
[50] Wu, Yunhui; Xue, Yuhao, Random hyperbolic surfaces of large genus have first eigenvalues greater than {\( \frac{3}{16}-\epsilon \)}, Geom. Funct. Anal.. Geometric and Functional Analysis, 32, 340-410 (2022) · Zbl 1487.32072 · doi:10.1007/s00039-022-00595-7
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