## 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

Zbl 0236.20032
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### References:

 [1] Sullivan, Publ. Math. I.H.E.S. 50 pp none– (none) [2] DOI: 10.1016/0021-8693(72)90058-0 · Zbl 0236.20032 [3] DOI: 10.1007/BF02761829 · Zbl 0431.22014 [4] Guivarc’h, Springer Lecture Notes 774 pp 176– (1980) [5] Goldsheid, Simplicity of the Liapunoff spectrum for products of random matrices 35 pp 309– (1987) [6] Glaner, Springer Lecture Notes 517 pp none– (1976) [7] Furstenberg, Proc. Symp. Pure Math. 6 pp 193– (1972) [8] Dixon, Free subgroups of linear groups 319 pp 45– (1973) · Zbl 0261.20039 [9] Cohen, Contemp. Math. Amer. Math. Soc. 50 pp none– (1986) [10] Chevalley, Théorie des Groupes de Lie (1968) [11] Bougerol, Products of Random Matrices with Applications to Schrödinger Operators (1986) · Zbl 0593.60044 [12] Margulis, Soviet Math. Dokl. 18 pp 847– (1977) [13] DOI: 10.1007/BF01388494 · Zbl 0551.20028 [14] Le Page, Springer Lecture Notes 928 pp 258– (1982) [15] Ledrappier, Israël J. Math. 50 pp none– (1985) [16] de la Harpe, Lecture Notes 1132 pp 248– [17] de la Harpe, L’Enseignement Math. 29 pp 129– (1983) [18] Guivarc’h, C.R.A.S. 304 pp 199– (1987) [19] DOI: 10.1007/BF02764859 · Zbl 0677.60007 [20] Guivarc’h, Bulletin Soc. Math. 101 pp 333– (1973) [21] DOI: 10.1007/BF02450281 · Zbl 0558.60009 [22] Zimmer, Ergodic theory and semi?simple groups (1984) · Zbl 0571.58015 [23] Weil, Grundlehren der Math Wiss. 144 (1967) [24] DOI: 10.1007/BF02760464 · Zbl 0415.28013
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