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Caractères de groupes algébriques sur Q et mesures invariantes sur les solénoïdes

Abstract : This thesis is divided in two parts in which the invariant probability measures on solenoids play a major role. The solenoids (that is a compact finite dimensional connected abelian group) are a natural generalization of the usual torus. In the first part, we will study the action of groups on a solenoid by affine transformation; we obtain a necessary and sufficient condition for the action of such a group to have the spectral gap property when the solenoid is provided with the Haar measure. In the second part we will study the trace and characters of algebraic groups over the field of rational numbers. The trace of a countable group are function of positive type on the group which are invariant under conjugation. The characters (that are the indecomposable traces in a certain way) are generalization of the usual characters of finite dimensional representations and intervene in the theory of operator algebra and in the study of invariant random subgroups. We begin with the classification of this characters in the case of unipotent groups. Then we extend this classification to general algebraic groups, using the study of the unipotent case et the establishment of the invariant measure on adelic solenoids.
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Submitted on : Tuesday, May 11, 2021 - 5:02:46 PM
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Camille Francini. Caractères de groupes algébriques sur Q et mesures invariantes sur les solénoïdes. Topologie algébrique [math.AT]. Université Rennes 1, 2020. Français. ⟨NNT : 2020REN1S078⟩. ⟨tel-03224450⟩



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