Spectra of elements in the group ring of

  • Alex Gamburd

    University of California at Santa Cruz, United States
  • Dmitry Jakobson

    McGill University, Montreal, Canada
  • Peter Sarnak

    Princeton University, United States

Abstract

We present a new method for establishing the “gap” property for finitely generated subgroups of , providing an elementary solution of Ruziewicz problem on as well as giving many new examples of finitely generated subgroups of with an explicit gap. The distribution of the eigenvalues of the elements of the group ring in the -th irreducible representation of is also studied. Numerical experiments indicate that for a generic (in measure) element of , the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for even) and the GSE spacing law (for odd) as ; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan) elements of 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, Dmitry Jakobson, Peter Sarnak, Spectra of elements in the group ring of . J. Eur. Math. Soc. 1 (1999), no. 1, pp. 51–85

DOI 10.1007/PL00011157