# Journal of the European Mathematical Society

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**Volume 1, Issue 1, 1999, pp. 51–85**

**DOI: 10.1007/PL00011157**

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

Alex Gamburd^{[1]}, Dmitry Jakobson

^{[2]}and Peter Sarnak

^{[3]}(1) Department of Mathematics, University of California at Santa Cruz, 1156 High Street, CA 95064, SANTA CRUZ, UNITED STATES

(2) Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West, QC H3A 2K6, MONTREAL, CANADA

(3) Department of Mathematics, Princeton University, Fine Hall, Washington Road, NJ 08544-1000, PRINCETON, UNITED STATES

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.

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Gamburd Alex, Jakobson Dmitry, Sarnak Peter: Spectra of elements in the group ring of SU(2). *J. Eur. Math. Soc.* 1 (1999), 51-85. doi: 10.1007/PL00011157