Pichot M., Removing chambers in Bruhat-Tits buildings

We introduce a family of countable groups constructed out of Euclidean buildings by “removing ” suitably chosen subsets of chambers. The corresponding simplicial complexes are called buildings with chambers missing. Let X be a CAT(0) space and let Γ be a countable group acting properly isometrically on X with X/Γ compact. A long-standing open problem is to understand if the alternative Γ is hyperbolic ↔ Γ contains a copy of Z2 holds under these assumptions (see [14]). The main idea of this paper is closely related to this problem. We start with a Euclidean building X, which we see as a space of “maximal rank”, and remove chambers from X. Since we are interested in constructing new groups, we remove chambers equivariantly with respect to the acting group Γ, and we assume that the rank of the building X is 2. Taking a universal cover of the resulting space, this gives rise to new groups (which in most cases are extensions of the original group Γ) and new CAT(0) spaces, called building with chambers missing. The “rank ” of these new spaces is a priori lower than that of the initial building. The basic reason for that, of course, is that all apartments containing the deleted chambers have disappeared. In some cases