Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T09:23:00.400Z Has data issue: false hasContentIssue false
  • English
  • Français

On Selberg’s eigenvalue conjecture for moduli spaces of abelian differentials

Part of: Discontinuous groups and automorphic forms Lie groups Deformations of analytic structures Riemann surfaces Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  30 October 2019

Michael Magee
Michael Magee*
Affiliation:
Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Rd, Durham DH1 3LE, UK email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

J.-C. Yoccoz proposed a natural extension of Selberg’s eigenvalue conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg’s $\frac{3}{16}$ theorem to moduli spaces of abelian differentials on surfaces of genus ${\geqslant}2$.

Keywords

abelian differentialsspectral gapcongruence covers

MSC classification

Primary: 11F72: Spectral theory; Selberg trace formula 22E40: Discrete subgroups of Lie groups 32G15: Moduli of Riemann surfaces, Teichmüller theory 30F60: Teichmüller theory 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 12 , December 2019 , pp. 2354 - 2398
Copyright
© The Author 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Athreya, J. S., Quantitative recurrence and large deviations for Teichmüller geodesic flow , Geom. Dedicata 119 (2006), 121–140.Google Scholar
Avila, A. and Gouëzel, S., Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow , Ann. of Math. (2) 178 (2013), 385–442.Google Scholar
Avila, A., Gouëzel, S. and Yoccoz, J.-C., Exponential mixing for the Teichmüller flow , Publ. Math. Inst. Hautes Études Sci. 104 (2006), 143–211.Google Scholar
Avila, A., Matheus, C. and Yoccoz, J.-C., Zorich conjecture for hyperelliptic Rauzy–Veech groups , Math. Ann. 370 (2018), 785–809.Google Scholar
Bargmann, V., Irreducible unitary representations of the Lorentz group , Ann. of Math. (2) 48 (1947), 568–640.Google Scholar
Bourgain, J. and Gamburd, A., Uniform expansion bounds for Cayley graphs of  SL2(𝔽p) , Ann. of Math. (2) 167 (2008), 625–642.Google Scholar
Bourgain, J., Gamburd, A. and Sarnak, P., Generalization of Selberg’s $\frac{3}{16}$ theorem and affine sieve, Acta Math. 207 (2011), 255–290.Google Scholar
Brooks, R., The spectral geometry of a tower of coverings , J. Differential Geom. 23 (1986), 97–107.Google Scholar
Burger, M., Estimation de petites valeurs propres du laplacien d’un revêtement de variétés riemanniennes compactes , C. R. Math. Acad. Sci. Paris Sér. I 302 (1986), 191–194.Google Scholar
Burger, M., Spectre du laplacien, graphes et topologie de Fell , Comment. Math. Helv. 63 (1988), 226–252.Google Scholar
Dolgopyat, D., On decay of correlations in Anosov flows , Ann. of Math. (2) 147 (1998), 357–390.Google Scholar
Eskin, A. and Mirzakhani, M., Invariant and stationary measures for the SL(2, R) action on Moduli space , Publ. Math. Inst. Hautes Études Sci. 127 (2018), 95–324.Google Scholar
Filip, S., Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle , Duke Math. J. 166 (2017), 657–706.Google Scholar
Gelbart, S. and Jacquet, H., A relation between automorphic representations of  GL(2) and GL(3) , Ann. Sci. Éc. Norm. Supér. (4) 11 (1978), 471–542.Google Scholar
Gowers, W. T., Quasirandom groups , Combin. Probab. Comput. 17 (2008), 363–387.Google Scholar
Gutiérrez-Romo, R., Classification of Rauzy–Veech groups: proof of the Zorich conjecture , Invent. Math. 215 (2019), 741–778.Google Scholar
Hejhal, D. and Sarnak, P., Some commentary on Atle Selberg’s mathematics , Bull. Amer. Math. Soc. (N.S.) 45 (2008), 485–487.Google Scholar
Iwaniec, H., The lowest eigenvalue for congruence groups , in Topics in geometry, Progress in Nonlinear Differential Equations and their Applications, vol. 20 (Birkhäuser Boston, Boston, MA, 1996), 203–212.Google Scholar
Každan, D. A., On the connection of the dual space of a group with the structure of its closed subgroups , Funktsional. Anal. i Prilozhen 1 (1967), 71–74.Google Scholar
Kelmer, D. and Silberman, L., A uniform spectral gap for congruence covers of a hyperbolic manifold , Amer. J. Math. 135 (2013), 1067–1085.Google Scholar
Kim, H. H., Functoriality for the exterior square of  GL4 and the symmetric fourth of  GL2 , J. Amer. Math. Soc. 16 (2003), 139–183.Google Scholar
Knapp, A. W., An overview based on examples , in Representation theory of semisimple groups, Princeton Landmarks in Mathematics, reprint of the 1986 original (Princeton University Press, Princeton, NJ, 2001).Google Scholar
Kontsevich, M. and Zorich, A., Connected components of the moduli spaces of Abelian differentials with prescribed singularities , Invent. Math. 153 (2003), 631–678.Google Scholar
Lubotzky, A., Phillips, R. and Sarnak, P., Ramanujan graphs , Combinatorica 8 (1988), 261–277.Google Scholar
Luo, W., Rudnick, Z. and Sarnak, P., On Selberg’s eigenvalue conjecture , Geom. Funct. Anal. 5 (1995), 387–401.Google Scholar
Maass, H., Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen , Math. Ann. 121 (1949), 141–183.Google Scholar
Magee, M., Quantitative spectral gap for thin groups of hyperbolic isometries , J. Eur. Math. Soc. (JEMS) 17 (2015), 151–187.Google Scholar
Magee, M., Oh, H. and Winter, D., Uniform congruence counting for Schottky semigroups in SL2(Z) , J. Reine Angew. Math. 753 (2019), 89–135.Google Scholar
Magee, M. and Rühr, R., Counting saddle connections in a homology class modulo q. With an Appendix by R. Gutiérrez-Romo , J. Mod. Dyn. 15 (2019), 237–262.Google Scholar
Margulis, G. A., Explicit constructions of expanders , Problemy Peredachi Informatsii 9 (1973), 71–80.Google Scholar
Marmi, S., Moussa, P. and Yoccoz, J.-C., The cohomological equation for Roth-type interval exchange maps , J. Amer. Math. Soc. 18 (2005), 823–872 (electronic).Google Scholar
Masur, H., Interval exchange transformations and measured foliations , Ann. of Math. (2) 115 (1982), 169–200.Google Scholar
Matthews, C. R., Vaserstein, L. N. and Weisfeiler, B., Congruence properties of Zariski-dense subgroups. I , Proc. Lond. Math. Soc. (3) 48 (1984), 514–532.Google Scholar
Nori, M. V., On subgroups of  GLn(F p) , Invent. Math. 88 (1987), 257–275.Google Scholar
Oh, H. and Winter, D., Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of  SL2(ℤ) , J. Amer. Math. Soc. 29 (2016), 1069–1115.Google Scholar
Ratner, M., The rate of mixing for geodesic and horocycle flows , Ergodic Theory Dynam. Systems 7 (1987), 267–288.Google Scholar
Rauzy, G., Échanges d’intervalles et transformations induites , Acta Arith. 34 (1979), 315–328.Google Scholar
Sarnak, P., Selberg’s eigenvalue conjecture , Notices Amer. Math. Soc. 42 (1995), 1272–1277.Google Scholar
Sarnak, P., Notes on the generalized Ramanujan conjectures , in Harmonic analysis, the trace formula, and Shimura varieties, Clay Mathematics Proceedings, vol. 4 (American Mathematical Society, Providence, RI, 2005), 659–685.Google Scholar
Sarnak, P. and Xue, X. X., Bounds for multiplicities of automorphic representations , Duke Math. J. 64 (1991), 207–227.Google Scholar
Seitz, G. M. and Zalesskii, A. E., On the minimal degrees of projective representations of the finite Chevalley groups. II , J. Algebra 158 (1993), 233–243.Google Scholar
Selberg, A., Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series , J. Indian Math. Soc. (N.S.) 20 (1956), 47–87.Google Scholar
Selberg, A., On the estimation of Fourier coefficients of modular forms , Proceedings of Symposia in Pure Mathematics, vol. 8 (American Mathematical Society, Providence, RI, 1965), 1–15.Google Scholar
Springer, T. A. and Steinberg, R., Conjugacy classes , in Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, NJ, 1968/69), Lecture Notes in Mathematics, vol. 131 (Springer, Berlin, 1970), 167–266.Google Scholar
Veech, W. A., Gauss measures for transformations on the space of interval exchange maps , Ann. of Math. (2) 115 (1982), 201–242.Google Scholar
Viana, M., Dynamics of interval exchange transformations and Teichmüller flows, Lecture Notes, IMPA (2008), http://w3.impa.br/∼viana/out/ietf.pdf.Google Scholar
Zorich, A., Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents , Ann. Inst. Fourier (Grenoble) 46 (1996), 325–370.Google Scholar
Zorich, A., How do the leaves of a closed 1-form wind around a surface? in Pseudoperiodic topology, American Mathematical Society Translations Series 2, vol. 197 (American Mathematical Society, Providence, RI, 1999), 135–178.Google Scholar
Zorich, A., Flat surfaces , in Frontiers in number theory, physics, and geometry I (Springer, Berlin, 2006), 437–583.Google Scholar