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Produits de matrices aléatoires et applications aux propriétés géométriques des sous-groupes du groupe linéaire. (Products of random matrices and applications to geometric properties of subgroups of linear groups). (French) Zbl 0715.60008

The aim of this study is to show how methods developed in the theory of dynamical systems and the theory of random matrices can be used to obtain deep results in group theory. Let k be a field of characteristic 0 and let \(\Gamma\) be a subgroup of the general linear group Gl(d,k). Introduce a probability measure \(\mu\) supported by a subset of \(\Gamma\). Let \(T_{\mu}\) denote the closed subsemigroup generated by \(\mu\). The elements of \(\Gamma\) may be regarded as k-valued matrices. This permits the author to use the multiplicative ergodic theorem by Oseledets to derive key properties of \(\Gamma\) and \(T_{\mu}\), in particular, precise growth rates of random products of \(\mu\)-distributed factors may be given. The potential benefits of combining the approach via random matrices with the more traditional methods from algebraic geometry are strongly stressed throughout the paper.
The first part is devoted to a new proof of the important theorem of J. Tits [J. Algebra 20, 250-270 (1972; Zbl 0236.20032)] stating that a subgroup of Gl(d,k) either has a non-abelian free subgroup or possesses a solvable subgroup of finite index. The argument invokes in an essential way the Oseledets theorem instead of the Zariski density used in the original proof by Tits.
In the second part, \(\Gamma\) is a subgroup of Sl(d,\({\mathfrak R})\). The author studies the set \(L_{\Gamma}\) of limit points in the Moore compactification of the symmetric space on which \(\Gamma\) is acting. It is shown, e.g., that the Hausdorff dimension of \(L_{\Gamma}\) is strictly positive. Also new results on the asymptotic behavior of the matrix products and their eigenvalues are obtained. The author proves, e.g., that (under some supplementary conditions) the eigenvalues of such products are (a.s.) real and simple.
Reviewer: G.Högnäs

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
20P05 Probabilistic methods in group theory
60G50 Sums of independent random variables; random walks

Citations:

Zbl 0236.20032
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References:

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