Selberg's 3/16 theorem is about a uniform spectral gap for all congruence subgroups of $SL_2(\mathbb{Z})$. For a finitely generated subgroup $\Gamma$ of $SL_2(\mathbb{Z})$ with critical exponent bigger than 1/2, Bourgain, Gamburd and Sarnak established a generalization of Selberg's theorem, using their expander results and the $L^2$-spectral theory. When the critical exponent of Gamma is at most 1/2, the $L^2$-theory is not applicable any more and the case remained open. I will discuss recent joint work with Dale Winter which resolves this issue by combining Dolgopyat's dynamical proof for exponential mixing of geodesic flow with the expansion machinery. As a consequence, we obtain a uniform resonance-free half plane for the resolvent of the Laplacian.
If time permits, I will also discuss a separate work with Michael Magee and Dale Winter, in which we have extended these methods in order to obtain spectral bounds for congruence Ruelle operators associated to random walks of a free semigroup arising from the study of continued fractions.
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