# Spectra of elements in the group ring of SU(2)

### Alex Gamburd

University of California at Santa Cruz, United States### Peter Sarnak

Princeton University, United States### Dmitry Jakobson

McGill University, Montreal, Canada

## Abstract

We present a new method for establishing the "gap" property for finitely generated subgroups of SU(2), providing an elementary solution of Ruziewicz problem on S2 as well as giving many new examples of finitely generated subgroups of SU(2) with an explicit gap. The distribution of the eigenvalues of the elements of the group ring R[SU(2)] in the N-th irreducible representation of SU(2) is also studied. Numerical experiments indicate that for a generic (in measure) element of R[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 NMX; we establish several results in this direction. For certain special "arithmetic" (or Ramanujan) elements of R[SU(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.

## Cite this article

Alex Gamburd, Peter Sarnak, Dmitry Jakobson, Spectra of elements in the group ring of SU(2). J. Eur. Math. Soc. 1 (1999), no. 1, pp. 51–85

DOI 10.1007/PL00011157