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Random walk on a building of type Ãr and brownian motion of the Weyl chamber

Bruno Schapira (2009)

Annales de l'I.H.P. Probabilités et statistiques

In this paper we study a random walk on an affine building of type Ãr, whose radial part, when suitably normalized, converges toward the brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane (Probab. Theory Related Fields89 (1991) 117–129). This extends also the link at the probabilistic level between riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol...

Receding polar regions of a spherical building and the center conjecture

Bernhard Mühlherr, Richard M. Weiss (2013)

Annales de l’institut Fourier

We introduce the notion of a polar region of a spherical building and use some simple observations about polar regions to give elementary proofs of various fundamental properties of root groups. We combine some of these observations with results of Timmesfeld, Balser and Lytchak to give a new proof of the center conjecture for convex chamber subcomplexes of thick spherical buildings.

Reflection loops of spaces with congruence and hyperbolic incidence structure

Alexander Kreuzer (2004)

Commentationes Mathematicae Universitatis Carolinae

In an absolute space ( P , 𝔏 , ≡ , α ) with congruence there are line reflections and point reflections. With the help of point reflections one can define in a natural way an addition + of points which is only associative if the product of three point reflection is a point reflection again. In general, for example for the case that ( P , 𝔏 , α ) is a linear space with hyperbolic incidence structure, the addition is not associative. ( P , + ) is a K-loop or a Bruck loop.

Réseaux de Coxeter-Davis et commensurateurs

Frédéric Haglund (1998)

Annales de l'institut Fourier

For each integer k ≥ 6 and each finite graph L , we construct a Coxeter group W and a non positively curved polygonal complex A on which W acts properly cocompactly, such that each polygon of A has k edges, and the link of each vertex of A is isomorphic to L . If L is a “generalized m -gon”, then A is a Tits building modelled on a reflection group of the hyperbolic plane. We give a condition on Aut ( L ) for Aut ( A ) to be non enumerable (which is satisfied if L is a thick classical generalized m -gon). On the other hand,...

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