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

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

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 michael.r.magee@durham.ac.uk
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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 

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