# 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(2`g`,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 `D` ∈ `A`^{×}. 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