Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow

Abstract

We consider the $\mathrm{SL}(2, \mathbb{R})$ action on moduli spaces of quadratic differentials. If $\mu$ is an $\mathrm{SL}(2, \mathbb{R})$-invariant probability measure, crucial information about the associated representation on $L^2(\mu)$ (and, in particular, fine asymptotics for decay of correlations of the diagonal action, the Teichmüller flow) is encoded in the part of the spectrum of the corresponding foliated hyperbolic Laplacian that lies in $\!(0,1/4)\!$ (which controls the contribution of the complementary series). Here we prove that the essential spectrum of an invariant algebraic measure is contained in $[1/4,\infty)$; i.e., for every $\delta\!>\!0$, there are only finitely many eigenvalues (counted with multiplicity) in $(0,1/4\!-\!\delta)$. In particular, all algebraic invariant measures have a spectral gap.

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Authors

Artur Avila

CNRS UMR 7586, Institut de Mathématiques de Jussieu, Paris, France
and
IMPA, Rio de Janeiro, Brazil

Sébastien Gouëzel

IRMAR, Université de Rennes 1, Rennes, France