Spectra of elements in the group ring of
Alex Gamburd, Dmitry Jakobson, Peter Sarnak
Source 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.
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