Commentarii Mathematici Helvetici

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Volume 85, Issue 3, 2010, pp. 647–704
DOI: 10.4171/CMH/207

Published online: 2010-05-23

The large sieve and random walks on left cosets of arithmetic groups

Florent Jouve[1]

(1) Université Paris-Sud 11, Orsay, France

Building on Kowalski’s work on random walks on the groups SL(n,ℤ) and Sp(2g,ℤ), we consider similar problems (we try to estimate the probability with which, after k steps, the matrix obtained has a characteristic polynomial with maximal Galois group or has no nonzero squares among its entries) for more general classes of sets: in GL(n,A), where A is a subring of ℚ containing ℤ that we specify, we perform a random walk on the set of matrices with fixed determinant DA×. We also investigate the case where the set involved is any of the two left cosets of the special orthogonal group SO(n,m)(ℤ) with respect to the spinorial kernel Ω(n,m)(ℤ).

Keywords: Random walks on arithmetic groups, Property (τ), large sieve, polynomials and orthogonal matrices over finite fields

Jouve Florent: The large sieve and random walks on left cosets of arithmetic groups. Comment. Math. Helv. 85 (2010), 647-704. doi: 10.4171/CMH/207