Jean Lécureux

Trees in Coxeter Complexes, automorphism groups of buildings and harmonic analysis: We prove that a Coxeter group is isometrically and equivariantly embedded into a finite product of homogeneous trees. In the affine case --and only in this case--, these trees are line: thus, we get a dichotomy between affine and non-affine Coxeter groups. At the level of building, this dichotomy gives rise to different behaviours of automorphism groups of buildings. For example, we prove that such a group admits a Gelfand pair if and only if the building is affine. We also use this embedding to prove that automorphism groups of buildings are amenable at infinity.