Right-angled buildings and combinatorial modulus on the boundary: Right-angled buildings and combinatorial modulus on the boundary

Abstract : The object of this thesis is to study the geometry of right-angled buildings. These spaces, defined by J. Tits, are singular spaces that can be seen as trees of higher dimension. The thesis is divided in two parts. In the first part, we describe how the notion of parallel residues allows to understand the action of a group on the building. As a corollary we recover that in Coxeter groups and in graph products intersections of parabolic subgroups are parabolic. In the second part, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings thanks to combinatorial tools. In particular, we exhibit some examples of buildings of dimension 3 and 4 whose boundary satisfy the combinatorial Loewner property.This property is a weak version of the Loewner property. This part is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D. Mostow. In the case of buildings of dimension 2, a lot of work has been done by M. Bourdon and H. Pajot. In particular, the Loewner property on the boundary permitted them to prove the quasi-isometry rigidity for some buildings of dimension 2.
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Antoine Clais. Right-angled buildings and combinatorial modulus on the boundary: Right-angled buildings and combinatorial modulus on the boundary. Group Theory [math.GR]. Université Lille 1, 2014. English. ⟨tel-01110542⟩

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