### Random matrices, magic squares and matching polynomials.

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We establish the spectral gap property for dense subgroups of SU$\left(d\right)$ $(d\ge 2)$, generated by finitely many elements with algebraic entries; this result was announced in [BG3]. The method of proof differs, in several crucial aspects, from that used in [BG] in the case of SU$\left(2\right)$.

We prove that the Cayley graphs of ${\mathrm{SL}}_{d}(\mathbb{Z}/{p}^{n}\mathbb{Z})$ are expanders with respect to the projection of any fixed elements in ${\mathrm{SL}}_{d}\left(\mathbb{Z}\right)$ generating a Zariski dense subgroup.

We present a new method for establishing the ‘‘gap” property for finitely generated subgroups of $SU\left(2\right)$, providing an elementary solution of Ruziewicz problem on ${S}^{2}$ as well as giving many new examples of finitely generated subgroups of $SU\left(2\right)$ with an explicit gap. The distribution of the eigenvalues of the elements of the group ring $\mathbf{R}[SU(2\left)\right]$ in the $N$-th irreducible representation of $SU\left(2\right)$ is also studied. Numerical experiments indicate that for a generic (in measure) element of $\mathbf{R}[SU(2\left)\right]$, the “unfolded” consecutive spacings...

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