Les travaux de J. Tits ont conduit à la classification complète des immeubles euclidiens de dimension supérieure ou égale à 3. L'ensemble de ces immeubles à isomorphisme près est dénombrable et paramétré par les corps locaux qui leur correspondent. Dans cet article nous nous intéressons aux immeubles triangulaires, qui sont euclidiens de dimension 2 et pour lesquelles une paramétrisation analogue est impossible. Nous construisons une lamination sur un espace topologique localement compact séparé, dont l'espace des feuilles est l'ensemble des immeubles triangulaires à isomorphisme près. On considère ainsi les immeubles triangulaires comme points d'un espace dont est une désingularisation naturelle. Nous établissons des résultats de chirurgie sur les immeubles triangulaires à données locales fixées. Ils entraînent par exemple que est topologiquement transitive. Nous montrons qu'un immeuble triangulaire générique au sens de Baire a un groupe d'automorphismes trivial et qu'il contient toutes les géométries locales possibles. L'espace des immeubles triangulaires à isomorphisme près est un nouvel exemple d'espace non commutatif.

We introduce and study a family of countable groups constructed from Euclidean buildings by "removing" suitably chosen subsets of chambers.In this paper X denotes a CAT(0) space and Γ is a countable group acting properly isometrically on X with X/Γ compact.The idea is the following. We start with a Euclidean building X of rank 2, which we see as a space of "maximal rank", and remove chambers from X equivariantly with respect to the acting group Γ. Taking a universal cover of the resulting space, this leads a new family of groups, which typically are extensions of the given group Γ, and a new class of CAT(0) spaces, which we call 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, the rank will decrease in a controlled way. One might expect, for example, that the least the proportion of removed chambers is, the closest the rank of these spaces is from the initial buildings. These groups and spaces are examples of objects "of intermediate rank" in the sense of [4]. Buildings X are viewed as spaces of maximal rank among their rank interpolating siblings (e.g. the triangle spaces in theà 2 case).Removing chambers in buildings leads to a rich supply of groups and spaces of intermediate rank on which the following alternative can be tested:Γ is hyperbolic ↔ Γ contains a copy of Z 2 see Section 6.B 3 in [14]. This problem is one of our original motivation, for this paper, and for "rank interpolation" in general.Let us now describe our main results.Buildings with chambers missing. A building with chambers missing consists of a simplicial complex X endowed with a free action of a countable group Γ with compact quotient, which satisfies certain axioms for chambers removal described in Section 1, Definition 1.1. As for usual Tits buildings, they are (in the non degenerate case) organized into types, according to the Coxeter diagram associated with them (which is inherited from the building they come from). Accordingly, we speak of spherical or Euclidean building with chambers missing when the diagram is finite or Euclidean. Euclidean buildings with chambers missing can be endowed with a natural piecewise linear metric, which is CAT(0) precisely when X is of dimension 2, see Section 1, if one chamber at least is missing (if no chamber is missing, i.e. if X is a Euclidean building, then the metric is always CAT(0) by well-known results of Bruhat and Tits). Henceforth

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On s'intéresse à la structure dynamique de l'espace E des immeubles triangulaires et on montre en particulier: -l'existence d'une infinité d'immeubles triangulaires quasi-périodiques non périodiques, -l'existence d'une infinité de composantes minimales non triviales dans E, c'est-à-dire de sous-laminations minimales de l'espace des immeubles pointés à isomorphisme pointé près, -l'existence d'une infinité de mesures transverses ergodiques quasi-invariantes non triviales à supports disjoints sur .Ceci reflète la variété des types de quasi-périodicité possibles pour les immeubles triangulaires et la complexité de l'espace E.Abstract This paper concerns the dynamical structure of the space of triangle buildings. It is proved that• there exist infinitely many triangular buildings which are quasi-periodic but not periodic, • there exist infinitely many minimal sublaminations of the space of pointed triangle buildings which are not reduced to a single leaf,

We consider models of random groups in which the typical group is of intermediate rank (in particular, it is not hyperbolic). These models are parallel to M. Gromov's well-known constructions and include for example a "density model" for groups of intermediate rank.The main novelty is the higher rank nature of the random groups. They are randomizations of certain families of lattices in algebraic groups (of rank 2) over local fields.This paper introduces models of random groups "of higher rank". The construction, basic properties, and applications are detailed in §1 to §7 below, which we now summarize.The construction (see §1) is rather general. If Γ ′ is a group which acts properly on a simply connected complex X ′ of dimension 2 with X ′ Γ ′ compact, and Γ ′′ ⊂ Γ ′ is a subgroup of "very large" finite index, then one can choose at random a family of Γ ′′ -orbits of 2-cells Y ⊂ X ′ inside X ′ . Then let X denote the universal cover of X ′′ ∶= X ′ ∖ Y . The random group Γ is the group of transformations of the Galois coveringThis construction leads to several distinct models of random groups including a "density model", following M. Gromov. The initial structural data (Γ ′ , X ′ ,...) for the model is called the deterministic data. The basic properties of Γ depend on the deterministic data.An idea of groups "of intermediate rank" was introduced in [2] in particular to address the following question, where X is CAT(0) and X Γ is compact:(This is the "periodic flat plane problem" which has been formulated in many places, see [12] for an early reference.) Since the assumption R 2 ↪ X is equivalent to X being non hyperbolic, the new models are relevant to the study of this question. We will see that in some cases (depending on the deterministic data, the density parameter, etc.) the answer is positive "generically", but that the precise relation between the two conditions "R 2 ↪ X" and "Z 2 ↪ Γ" remains mysterious even for random groups associated with lattices in PSL 3 .Before turning to these models let us discuss briefly Gromov's original construction of random groups and the density model introduced in [14] (see also [12, §6], [13] or [15]).