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.

We give combinatorial models for non-spherical, generic, smooth, complex representations of the group $G=\mathrm{PGL}(2,F)$, where $F$ is a non-Archimedean locally compact field. More precisely we carry on studying the graphs ${\left({\tilde{X}}_k\right)}_{k\ge 0}$ defined in a previous work. We show that such representations may be obtained as quotients of the cohomology of a graph ${\tilde{X}}_k$, for a suitable integer $k$, or equivalently as subspaces of the space of discrete harmonic cochains on such a graph. Moreover, for supercuspidal representations, these models...

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,...