Magee–Avila–Gouëzel conjecture for moduli spaces of abelian differentials

Let $\mathcal{M}$ be a connected component of a stratum of abelian differentials, let $\mathcal{M}(q)$ be its level-$q$ lift for $q\geq 2$, and let $\nu_{\mathcal{M}(q)}$ be the corresponding finite invariant measure. Let $\Delta_{\mathcal{M}(q)}$ be the foliated Laplacian and let $\lambda_1(\mathcal{M}(q))$ be the infimum of its non-zero spectrum. Magee–Avila–Gouëzel conjecture. For all $q\geq 2$ and every connected component $\mathcal{M}$ of a stratum, both of the following hold:

1. $\lambda_1(\mathcal{M}(q))\geq\frac{1}{4}$.
2. The measure on the unitary dual of $\operatorname{SL}_2(\mathbf{R})$ giving the decomposition of

$$L^2(\mathcal{M}(q),\nu_{\mathcal{M}(q)})$$

is supported away from complementary-series representations.

This conjecture extends Selberg's eigenvalue conjecture to moduli spaces of abelian differentials and was suggested by Avila and Gouëzel. The paper proves an approximation to it; it remains open, and it is not known in general whether the relevant foliated Laplacians have any non-zero eigenvalues.