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Automorphismes et compactifications d’immeubles : moyennabilité et action sur le bord

Abstract : The object of this thesis is the study, from different point of views, of automorphism groups of buildings. One of its objectives is to highlight the differences as well as the analogies between affine and non-affine buildings. In order to support this dichotomy, we prove that automorphism groups of non-affine buildings never have a Gelfand pair, contrarily to affine buildings.In the other direction, the analogy between affine and non-affine buildings is supported by the new construction of a combinatorial boundary of a building. In the affine case, this boundary is in fact the polyhedral boundary. We connect the construction of this boundary to other compactifications, such as the Busemann compactification of the graph of chambers. The combinatorial compactification is also isomorphic to the group-theoretic compactification, which embeds the set of chambers into the set of closed subgroups of the automorphism group. We also connect the combinatorial boundary to another space, which generalises a construction of F. Karpelevic for symmetric spaces : the refined boundary of a CAT(0) space.We prove that the maximal amenable subgroups of the automorphism group are, up to finite index, parametrised by the points of the boundary. Finally, we prove that the action of the automorphism group of a locally finite building on its combinatorial boundary is amenable, thus providing resolutions in bounded cohomology and boundary maps. This also gives a new proof that these groups satisfy the Novikov conjecture.
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Submitted on : Monday, April 23, 2012 - 12:42:23 PM
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Jean Lécureux. Automorphismes et compactifications d’immeubles : moyennabilité et action sur le bord. Mathématiques générales [math.GM]. Université Claude Bernard - Lyon I, 2009. Français. ⟨NNT : 2009LYO10261⟩. ⟨tel-00463411v2⟩



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