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...

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.

For each integer $k\ge 6$ 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 $\mathrm{Aut}\left(L\right)$ for $\mathrm{Aut}\left(A\right)$ to be non enumerable (which is satisfied if $L$ is a thick classical generalized $m$-gon). On the other hand,...