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We establish a spectral gap for resonances of the Laplacian of random Schottky surfaces, which is optimal according to a conjecture of Jakobson and Naud.
Samuel J. Patterson, The limit set of a Fuchsian group , Acta Math. 136
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Noga Alon, Eigenvalues and expanders , Combinatorica 6
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Laurent Guillopé and Maciej Zworski, Upper bounds on the number of resonances for non-compact Riemann surfaces , J. Funct. Anal. 129
1995
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by same author, Scattering asymptotics for Riemann surfaces , Ann. Math. (2) 145
1997
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Jack Button, All Fuchsian Schottky groups are classical Schottky groups , The Epstein Birthday Schrift Dedicated to David Epstein on the Occasion of his 60th Birthday, Warwick: University of Warwick, Institute of Mathematics, 1998, pp. 117–125
1998
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Artur Buchholz, Norm of convolution by operator-valued functions on free groups , Proceedings of the American Mathematical Society 127
1999
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Alex Gamburd, On the spectral gap for infinite index “congruence” subgroups of SL ( 2 , ℤ ) \mathrm{SL}(2,\mathbb{Z}) , Isr. J. Math. 127
2002
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Joel Friedman, Relative expanders or weakly relatively Ramanujan graphs. , Duke Math. J. 118
2003
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Robert Brooks and Eran Makover, Random construction of Riemann surfaces , J. Differ. Geom. 68
2004
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Uffe Haagerup and Steen Thorbjørnsen, A new application of random matrices: Ext ( C red ∗ ( F 2 ) ) \operatorname{Ext}(C_{\text{red}}^{*}(F_{2})) is not a group , Ann. Math. (2) 162
2005
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Frédéric Naud, Expanding maps on Cantor sets and analytic continuation of zeta functions , Ann. Sci. Éc. Norm. Supér. (4) 38
2005
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Hanne Schultz, Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases. , Probab. Theory Relat. Fields 131
2005
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M. Capitaine and C. Donati-Martin, Strong asymptotic freeness for Wigner and Wishart matrices , Indiana Univ. Math. J. 56
2007
Cited alongside, same era.
by same author, A proof of Alon’s second eigenvalue conjecture and related problems , Mem. Am. Math. Soc., vol. 910, Providence, RI: American Mathematical Society (AMS), 2008 (English)
2008
Cited alongside, same era.
Mikael de la Salle, Strong Haagerup inequalities with operator coefficients , J. Funct. Anal. 257
2009
Cited alongside, same era.
Jean Bourgain, Alex Gamburd, and Peter Sarnak, Generalization of Selberg’s 3 16 \frac{3}{16} theorem and affine sieve , Acta Math. 207
2011
Cited alongside, same era.
Dmitry Jakobson and Frédéric Naud, On the critical line of convex co-compact hyperbolic surfaces , Geom. Funct. Anal. 22
2012
Cited alongside, same era.
by same author, Spectral gaps without the pressure condition , Ann. Math. (2) 187
2018
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Charles Bordenave and Benoît Collins, Eigenvalues of random lifts and polynomials of random permutation matrices , Ann. Math. (2) 190
2019
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Michael Magee, Hee Oh, and Dale Winter, Uniform congruence counting for Schottky semigroups in SL ( 2 , ℤ ) \mathrm{SL}(2,\mathbb{Z}) , with appendix by Jean Bourgain, Alex Kontorovich, and Michael Magee , J. Reine Angew. Math. 2019
2019
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Charles Bordenave, A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts , Ann. Sci. Éc. Norm. Supér. (4) 53
2020
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Dmitry Jakobson, Frédéric Naud, and Louis Soares, Large degree covers and sharp resonances of hyperbolic surfaces , Ann. Inst. Fourier 70
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Greg W. Anderson, Convergence of the largest singular value of a polynomial in independent Wigner matrices , Ann. Probab. 41
2013
Cited alongside, same era.
Maryam Mirzakhani, Growth of Weil-Petersson volumes and random hyperbolic surface of large genus , J. Differ. Geom. 94
2013
Cited alongside, same era.
Benoît Collins and Camille Male, The strong asymptotic freeness of Haar and deterministic matrices , Ann. Sci. Éc. Norm. Supér. (4) 47
2014
Cited alongside, same era.
Peter Sarnak, Notes on thin matrix groups , Thin groups and superstrong approximation. Selected papers based on the presentations at the workshop held at MSRI, Berkeley, CA, USA, February 6–10, 2012, Cambridge: Cambridge University Press, 2014, pp. 343–362 (English)
2014
Cited alongside, same era.
Hee Oh and Dale Winter, Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of SL ( 2 , ℤ ) \mathrm{SL}(2,\mathbb{Z}) , J. Am. Math. Soc. 29
2015
Cited alongside, same era.
Doron Puder, Expansion of random graphs: new proofs, new results , Invent. Math. 201
2015
Cited alongside, same era.
David Borthwick, Spectral theory of infinite-area hyperbolic surfaces , second ed., Prog. Math., vol. 318, Birkhäuser/Springer, [Cham], 2016
2016
Cited alongside, same era.
2020
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Michael Magee and Frédéric Naud, Explicit spectral gaps for random covers of Riemann surfaces , Publications Mathématiques. Institut de Hautes Études Scientifiques 132
2020
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Clifford Gilmore, Etienne Le Masson, Tuomas Sahlsten, and Joe Thomas, Short geodesic loops and L p L^{p} norms of eigenfunctions on large genus random surfaces , Geom. Funct. Anal. 31
2021
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Michael Magee, Frédéric Naud, and Doron Puder, A random cover of a compact hyperbolic surface has relative spectral gap 3 16 − ε \frac{3}{16}-\varepsilon , Geom. Funct. Anal. 32
2022
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Laura Monk, Benjamini-Schramm convergence and spectra of random hyperbolic surfaces of high genus , Anal. PDE 15
2022
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Yunhui Wu and Yuhao Xue, Random hyperbolic surfaces of large genus have first eigenvalues greater than 3 16 − ϵ \frac{3}{16}-\epsilon , Geom. Funct. Anal. 32
2022
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Irving Calderón and Michael Magee, Explicit spectral gaps for Schottky groups contained in SL ( 2 , ℤ ) \mathrm{SL}(2,\mathbb{Z}) , Preliminary version (2023)
2023
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Will Hide and Michael Magee, Near optimal spectral gaps for hyperbolic surfaces , Ann. Math. (2) 198
2023
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Michael Magee and Doron Puder, The asymptotic statistics of random covering surfaces , Forum Math. Pi 11
2023
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Michael Magee and Joe Thomas, Strongly convergent unitary representations of right-angled artin groups , 2023
2023
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Michael Lipnowski and Alex Wright, Towards optimal spectral gaps in large genus , Ann. Probab. 52
2024
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