  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.
