Explicit spectral gaps for random covers of Riemann surfaces

Magee, Michael; Naud, Frédéric

doi:10.1007/s10240-020-00118-w

Article

Explicit spectral gaps for random covers of Riemann surfaces

Magee, Michael ; Naud, Frédéric

Publications Mathématiques de l'IHÉS, Tome 132 (2020), pp. 137-179

Résumé

We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X = Γ ∖ 𝐇$ . Let $δ$ be the Hausdorff dimension of the limit set of $Γ$ . We say that a resonance of $X_{n}$ is new if it is not a resonance of $X$ , and similarly define new eigenvalues of the Laplacian.

We prove that for any $ϵ > 0$ and $H > 0$ , with probability tending to 1 as $n \to \infty$ , there are no new resonances $s = σ + i t$ of $X_{n}$ with $σ \in [\frac{3}{4} δ + ϵ, δ]$ and $t \in [- H, H]$ . This implies in the case of $δ > \frac{1}{2}$ that there is an explicit interval where there are no new eigenvalues of the Laplacian on $X_{n}$ . By combining these results with a deterministic ‘high frequency’ resonance-free strip result, we obtain the corollary that there is an $η = η (X)$ such that with probability $\to 1$ as $n \to \infty$ , there are no new resonances of $X_{n}$ in the region ${s : Re (s) > δ - η}$ .

Reçu le : 2019-07-04
Accepté le : 2020-06-10
Première publication : 2020-06-25
Publié le : 2020-12-28

MR Zbl

DOI : 10.1007/s10240-020-00118-w

@article{PMIHES_2020__132__137_0,
     author = {Magee, Michael and Naud, Fr\'ed\'eric},
     title = {Explicit spectral gaps for random covers of {Riemann} surfaces},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {137--179},
     year = {2020},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {132},
     doi = {10.1007/s10240-020-00118-w},
     mrnumber = {4179833},
     zbl = {1508.58008},
     language = {en},
     url = {https://www.numdam.org/articles/10.1007/s10240-020-00118-w/}
}

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Magee, Michael; Naud, Frédéric. Explicit spectral gaps for random covers of Riemann surfaces. Publications Mathématiques de l'IHÉS, Tome 132 (2020), pp. 137-179. doi: 10.1007/s10240-020-00118-w

Bibliographie
Cité par

[Alo86] Alon, N. Eigenvalues and expanders, Combinatorica, Volume 6 (1986), pp. 83-96 Theory of computing (Singer Island, Fla., 1984) | MR | Zbl | DOI

[BC19] Bordenave, C.; Collins, B. Eigenvalues of random lifts and polynomials of random permutation matrices, Ann. of Math. (2), Volume 190 (2019), pp. 811-875 | MR | DOI | Zbl

[BD17] Bourgain, J.; Dyatlov, S. Fourier dimension and spectral gaps for hyperbolic surfaces, Geom. Funct. Anal., Volume 27 (2017), pp. 744-771 | MR | Zbl | DOI

[BD18] Bourgain, J.; Dyatlov, S. Spectral gaps without the pressure condition, Ann. Math., Volume 187 (2018), pp. 825-867 | MR | Zbl | DOI

[BGS11] Bourgain, J.; Gamburd, A.; Sarnak, P. Generalization of Selberg’s $\frac{3}{16}$ theorem and affine sieve, Acta Math., Volume 207 (2011), pp. 255-290 | MR | Zbl | DOI

[BM04] Brooks, R.; Makover, E. Random construction of Riemann surfaces, J. Differ. Geom., Volume 68 (2004), pp. 121-157 | MR | Zbl | DOI

[BMM17] Ballmann, W.; Matthiesen, H.; Mondal, S. Small eigenvalues of surfaces of finite type, Compos. Math., Volume 153 (2017), pp. 1747-1768 | MR | Zbl | DOI

[Bol88] Bollobás, B. The isoperimetric number of random regular graphs, Eur. J. Comb., Volume 9 (1988), pp. 241-244 | MR | Zbl | DOI

[Bor16] Borthwick, D. Spectral Theory of Infinite-Area Hyperbolic Surfaces, 318, Springer, Cham, 2016 | Zbl | MR

[Bow79] Bowen, R. Hausdorff dimension of quasicircles, Publ. Math. IHÉS, Volume 50 (1979), pp. 11-25 | MR | Zbl | Numdam | DOI

[Bro86] Brooks, R. The spectral geometry of a tower of coverings, J. Differ. Geom., Volume 23 (1986), pp. 97-107 | MR | Zbl | DOI

[BS87] Broder, A.; Shamir, E. On the second eigenvalue of random regular graphs, The 28th Annual Symposium on Foundations of Computer Science, 1987, pp. 286-294 | DOI

[Bur88] Burger, M. Spectre du laplacien, graphes et topologie de Fell, Comment. Math. Helv., Volume 63 (1988), pp. 226-252 | MR | Zbl | DOI

[But98] Button, J. All Fuchsian Schottky groups are classical Schottky groups, The Epstein Birthday Schrift, 1, Geom. Topol. Publ., Coventry, 1998, pp. 117-125 | MR | Zbl

[BV05] Baladi, V.; Vallée, B. Euclidean algorithms are Gaussian, J. Number Theory, Volume 110 (2005), pp. 331-386 | MR | Zbl | DOI

[DJ18] Dyatlov, S.; Jin, L. Dolgopyat’s method and the fractal uncertainty principle, Anal. PDE, Volume 11 (2018), pp. 1457-1485 | MR | Zbl | DOI

[Dol98] Dolgopyat, D. On decay of correlations in Anosov flows, Ann. Math., Volume 147 (1998), pp. 357-390 | MR | Zbl | DOI

[Dya19] Dyatlov, S. An introduction to fractal uncertainty principle, J. Math. Phys., Volume 60 (2019) | MR | DOI | Zbl

[DZ17] Dyatlov, S.; Zworski, M. Fractal uncertainty for transfer operators, Int. Math. Res. Not., Volume 2020 (2020), pp. 781-812 | MR | DOI | Zbl

[FP17] Fedosova, K.; Pohl, A. Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy, Sel. Math., Volume 26 (2020), p. 9 | MR | DOI | Zbl

[Fri03] Friedman, J. Relative expanders or weakly relatively Ramanujan graphs, Duke Math. J., Volume 118 (2003), pp. 19-35 | MR | DOI | Zbl

[Fri08] Friedman, J. A proof of Alon’s second eigenvalue conjecture and related problems, Mem. Am. Math. Soc., Volume 195 (2008), p. 910 (viii+100) | MR | Zbl

[Gam02] Gamburd, A. On the spectral gap for infinite index “congruence” subgroups of ${SL}_{2} (𝐙)$ , Isr. J. Math., Volume 127 (2002), pp. 157-200 | MR | Zbl | DOI

[Gam06] Gamburd, A. Poisson-Dirichlet distribution for random Belyi surfaces, Ann. Probab., Volume 34 (2006), pp. 1827-1848 | MR | Zbl | DOI

[GLZ04] Guillopé, L.; Lin, K. K.; Zworski, M. The Selberg zeta function for convex co-compact Schottky groups, Commun. Math. Phys., Volume 245 (2004), pp. 149-176 | MR | Zbl | DOI

[GN09] Guillarmou, C.; Naud, F. Wave decay on convex co-compact hyperbolic manifolds, Commun. Math. Phys., Volume 287 (2009), pp. 489-511 | MR | Zbl | DOI

[Gui92] Guillopé, L. Fonctions zêta de Selberg et surfaces de géométrie finie, Zeta Functions in Geometry, Volume 21 (1992), pp. 33-70 | Zbl | MR | DOI

[GZ95] Guillopé, L.; Zworski, M. Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Funct. Anal., Volume 129 (1995), pp. 364-389 | MR | Zbl | DOI

[JN16] Jakobson, D.; Naud, F. Resonances and density bounds for convex co-compact congruence subgroups of $S L_{2} (𝐙)$ , Isr. J. Math., Volume 213 (2016), pp. 443-473 | Zbl | MR | DOI

[JNS19] Jakobson, D.; Naud, F.; Soares, L. Large covers and sharp resonances of hyperbolic surfaces, Ann. Inst. Fourier, Volume 70 (2020), pp. 523-596 | MR | DOI | Zbl

[JZ17] Jin, L.; Zhang, R. Fractal uncertainty principle with explicit exponent, Math. Ann., Volume 376 (2020), pp. 1031-1057 | MR | DOI | Zbl

[Liv95] Liverani, C. Decay of correlations, Ann. Math., Volume 142 (1995), pp. 239-301 | MR | Zbl | DOI

[LP81] Lax, P. D.; Phillips, R. S. The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, Functional Analysis and Approximation, Volume 60 (1981), pp. 373-383 | MR | Zbl | DOI

[LPS88] Lubotzky, A.; Phillips, R.; Sarnak, P. Ramanujan graphs, Combinatorica, Volume 8 (1988), pp. 261-277 | MR | Zbl | DOI

[Mag15] Magee, M. Quantitative spectral gap for thin groups of hyperbolic isometries, J. Eur. Math. Soc., Volume 17 (2015), pp. 151-187 | MR | Zbl | DOI

[MM87] Mazzeo, R. R.; Melrose, R. B. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., Volume 75 (1987), pp. 260-310 | MR | Zbl | DOI

[MOW17] M. Magee, H. Oh and D. Winter, Uniform congruence counting for Schottky semigroups in ${SL}_{2} (𝐙)$ ), J. Reine Angew. Math. (2017), with appendix by J. Bourgain, a. Kontorovich, and M. Magee. | MR

[Nau05a] Naud, F. Expanding maps on Cantor sets and analytic continuation of zeta functions, Ann. Sci. Éc. Norm. Supér., Volume 38 (2005), pp. 116-153 | MR | Zbl | Numdam | DOI

[Nau05b] Naud, F. Precise asymptotics of the length spectrum for finite-geometry Riemann surfaces, Int. Math. Res. Not., Volume 5 (2005), pp. 299-310 | MR | Zbl | DOI

[Nau14] Naud, F. Density and location of resonances for convex co-compact hyperbolic surfaces, Invent. Math., Volume 195 (2014), pp. 723-750 | MR | Zbl | DOI

[Nil91] Nilli, A. On the second eigenvalue of a graph, Discrete Math., Volume 91 (1991), pp. 207-210 | MR | Zbl | DOI

[OW16] Oh, H.; Winter, D. Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of ${SL}_{2} (𝐙)$ , J. Am. Math. Soc., Volume 29 (2016), pp. 1069-1115 | Zbl | MR | DOI

[Pat76] Patterson, S. J. The limit set of a Fuchsian group, Acta Math., Volume 136 (1976), pp. 241-273 | MR | Zbl | DOI

[PP90] Parry, W.; Pollicott, M. Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, Volume 187–188 (1990), p. 268 | MR | Zbl | Numdam

[PP01] Patterson, S. J.; Perry, P. A. The divisor of Selberg’s zeta function for Kleinian groups, Duke Math. J., Volume 106 (2001), pp. 321-390 (Appendix A by Charles Epstein) | MR | Zbl | DOI

[PP15] Puder, D.; Parzanchevski, O. Measure preserving words are primitive, J. Am. Math. Soc., Volume 28 (2015), pp. 63-97 | MR | Zbl | DOI

[Pud15] Puder, D. Expansion of random graphs: new proofs, new results, Invent. Math., Volume 201 (2015), pp. 845-908 | MR | Zbl | DOI

[Sel65] Selberg, A. On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., VIII, Amer. Math. Soc., Providence, 1965, pp. 1-15 | MR | Zbl

[VZ82] Venkov, A. 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), pp. 1150-1158 (1343) | MR | Zbl

[Zwo17] Zworski, M. Mathematical study of scattering resonances, Bull. Math. Sci., Volume 7 (2017), pp. 1-85 | MR | Zbl | DOI

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