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.
@article{10_4007_annals_2013_178_2_1,
author = {Artur Avila and S\'ebastien Gou\"ezel},
title = {Small eigenvalues of the {Laplacian} for algebraic measures in moduli space, and mixing properties of the {Teichm\"uller} flow},
journal = {Annals of mathematics},
pages = {385--442},
year = {2013},
volume = {178},
number = {2},
doi = {10.4007/annals.2013.178.2.1},
mrnumber = {3071503},
zbl = {06203671},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.2.1/}
}
TY - JOUR AU - Artur Avila AU - Sébastien Gouëzel TI - Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow JO - Annals of mathematics PY - 2013 SP - 385 EP - 442 VL - 178 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.2.1/ DO - 10.4007/annals.2013.178.2.1 LA - en ID - 10_4007_annals_2013_178_2_1 ER -
%0 Journal Article %A Artur Avila %A Sébastien Gouëzel %T Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow %J Annals of mathematics %D 2013 %P 385-442 %V 178 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.2.1/ %R 10.4007/annals.2013.178.2.1 %G en %F 10_4007_annals_2013_178_2_1
Artur Avila; Sébastien Gouëzel. Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow. Annals of mathematics, Tome 178 (2013) no. 2, pp. 385-442. doi: 10.4007/annals.2013.178.2.1
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