Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichm\"uller flow

We consider the $SL(2,R)$ action on moduli spaces of quadratic differentials. If $\mu$ is an $SL(2,R)$-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 Teichm\"uller 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.