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Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow. (English) Zbl 1287.58016
The Teichmüller geodesic flow on the moduli space of quadratic differentials is an important naturally occurring example of a non-uniformly hyperbolic dynamical system, and has been the object of a significant amount of recent study. In particular, the mixing properties of the flow, and the related questions of the spectral properties of the larger $$\text{SL}(2, \mathbb R)$$ action on the space of quadratic differentials have been studied closely by, among others, Avila-Gouëzel-Yoccoz, Avila-Resende, and many others. In this paper, the authors make progress towards answering a question of Yoccoz, who asked for which $$\text{SL}(2, \mathbb R)$$-invariant measures does the analog of the Ramanujan conjecture hold- that is, when does the spectrum for the foliated hyperbolic Laplacian associated to the invariant measure have no eigenvalues between 0 and $$1/4$$? The main result of this paper is that for all algebraic invariant measures (that is, invariant measures supported on algebraic submanifolds of the moduli space), for any $$\delta >0$$, there are only finitely many eigenvalues in $$(0, \frac 1 4 - \delta)$$. Combining this with recent results of Eskin-Mirzakhani, which show that all $$\text{SL}(2, \mathbb R)$$ measures are algebraic, this result shows that the essential part of the spectrum for any $$\text{SL}(2, \mathbb R)$$-invariant measure is in $$[1/4, \infty)$$. In particular this shows that exponential mixing of the Teichmüller flow holds for any $$\text{SL}(2, \mathbb R)$$-invariant measure.

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
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