Spectra of elements in the group ring of $\mathrm{SU}(2)$

Abstract

We present a new method for establishing the “gap” property for finitely generated subgroups of $\mathrm{SU}(2)$, providing an elementary solution of Ruziewicz problem on $S^2$ as well as giving many new examples of finitely generated subgroups of $\mathrm{SU}(2)$ with an explicit gap. The distribution of the eigenvalues of the elements of the group ring $\mathbf{R}[\mathrm{SU}(2)]$ in the $N$-th irreducible representation of $\mathrm{SU}(2)$ is also studied. Numerical experiments indicate that for a generic (in measure) element of $\mathbf{R}[\mathrm{SU}(2)]$, the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for $N$ even) and the GSE spacing law (for $N$ odd) as $N\to \infty$; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan) elements of $\mathbf{R}[\mathrm{SU}(2)]$ the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.