Spectra of elements in the group ring of SU(2) |
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Authors: | Alex Gamburd Dmitry Jakobson Peter Sarnak |
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Institution: | (1) Department of Mathematics, Princeton University, Princeton, NJ 08544, USA, US;(2) IAS, School of Mathematics, Princeton, NJ 08540 and Mathematics, 253-37, Caltech, Pasadena, CA 91125, USA, US |
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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 RSU(2)] in the N-th irreducible representation of SU(2) is also studied. Numerical experiments indicate that for a generic
(in measure) element of RSU(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→∞; we establish several results in this direction. For certain special “arithmetic”
(or Ramanujan) elements of RSU(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide
a sharp estimate in that direction.
Received June 1, 1998 / final version received September 8, 1998 |
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Keywords: | Mathematics Subject Classification (1991): 11 22E45 42Axx 54H15 81Q50 |
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