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

### Florent Jouve

Université Paris-Sud 11, Orsay, France

## Abstract

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 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).

## Cite this article

Florent Jouve, The large sieve and random walks on left cosets of arithmetic groups. Comment. Math. Helv. 85 (2010), no. 3, pp. 647–704

DOI 10.4171/CMH/207