Application of random walks to the study of subgroups of linear groups

Abstract : In this thesis, we use and contribute to the theory of random matrix products in order to study generic properties of elements and subgroups of linear groups. Our first result gives a probabilistic version of the Tits alternative : we show that two independent random walks M_n and M'_n on a non virtually solvable finitely generated linear group will eventually generate a non abelian free subgroup. This answers a question of Guivarc'h and Gilman, Miasnikov and Osin. We show in fact that the probability that M_n and M'_n do not generate a free subgroup decreases exponentially fast to zero. Our methods rely deeply on random matrix products theory. During the proof we give some new limit theorems concerning this theory, some of them will be the generalization of known results for matrices taking value in archimedean fields to arbitrary local fields, others will be new even over R. For example, we show that under natural assumptions on the random walk, the K-parts of M_n in the KAK decomposition become asymptotically independent with exponential speed. Next, we use these properties to study the transience of algebraic subvarieties in algebraic groups. One of our results can be formulated as follows: let H be a non elementary subgroup of SL_2(R), a probability measure with an exponential moment whose support generates H, then for every proper algebraic subvariety V of SL_2(R), the probability that the random walk lies in V decreases exponentially fast to zero. This shows that every proper algebraic subvariety is transient for the random walk. We generalize this result to the case where the support of the probability measure generates a Zariski dense subgroup of the real points of an algebraic group defined and split over R. These results share common flavor with recent works of Kowalski and Rivin
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https://tel.archives-ouvertes.fr/tel-00601922
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Submitted on : Tuesday, June 21, 2011 - 9:08:12 AM
Last modification on : Saturday, April 20, 2019 - 1:16:28 AM
Long-term archiving on : Thursday, September 22, 2011 - 2:21:16 AM

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Richard Aoun. Application of random walks to the study of subgroups of linear groups. General Mathematics [math.GM]. Université Paris Sud - Paris XI, 2011. English. ⟨NNT : 2011PA112066⟩. ⟨tel-00601922⟩

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