For any prime p and S a p-group isomorphic to a Sylow p-subgroup of a rank $2$ simple group of Lie type in characteristic p, we determine all saturated fusion systems supported on S up to isomorphism.

We solve a fundamental question posed in Frohardt’s 1988 paper [6] on finite $2$-groups with Kantor familes, by showing that finite groups K with a Kantor family $(\mathcal {F},\mathcal {F}^*)$ having distinct members $A, B \in \mathcal {F}$ such that $A^* \cap B^*$ is a central subgroup of K and the quotient $K/(A^* \cap B^*)$ is abelian cannot exist if the center of K has exponent $4$ and the members of $\mathcal {F}$ are elementary abelian. Then we give a short geometrical proof of a recent result of Ott which says that finite skew translation quadrangles of even order $(t,t)$ (where t is not a square) are always translation generalized quadrangles. This is a consequence of a complete classification of finite cyclic skew translation quadrangles of order $(t,t)$ that we carry out in the present paper.

To each automorphism of a spherical building, there is a naturally associated opposition diagram, which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber, then the automorphism is called domestic. In this paper, we give the complete classification of domestic automorphisms of split spherical buildings of types $\mathsf {E}_6$ , $\mathsf {F}_4$ , and $\mathsf {G}_2$ . Moreover, for all split spherical buildings of exceptional type, we classify (i) the domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most two distinguished orbits encircled. Our results provide unexpected characterizations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.

Soient K un corps discrètement valué et hensélien, ${\mathcal {O}}$ son anneau d’entiers supposé excellent, $\kappa $ son corps résiduel supposé parfait et G un K-groupe quasi-réductif, c’est-à-dire lisse, affine, connexe et à radical unipotent déployé trivial. On construit l’immeuble de Bruhat-Tits ${\mathcal {I}}(G, K)$ pour $G(K)$ de façon canonique, améliorant les constructions moins canoniques de M. Solleveld sur les corps locaux, et l’on associe un ${\mathcal {O}}$ -modèle en groupes ${\mathcal {G}}_{\Omega }$ de G à chaque partie non vide et bornée $\Omega $ contenue dans un appartement de ${\mathcal {I}}(G,K)$ . On montre que les groupes parahoriques ${\mathcal {G}}_{\textbf {f}}$ attachés aux facettes peuvent être caractérisés en fonction de la géométrie de leurs grassmanniennes affines, ainsi que dans la thèse de T. Richarz. Ces résultats sont appliqués ailleurs à l’étude des grassmanniennes affines tordues entières.

Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We characterize a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular, many of the exceptional Lie incidence geometries occur. In an appendix, we extend our result to the locally disconnected case and discuss the locally disconnected case of some other well-known characterizations.

Masures are generalizations of Bruhat–Tits buildings. They were introduced by Gaussent and Rousseau to study Kac–Moody groups over ultrametric fields that generalize reductive groups. Rousseau gave an axiomatic definition of these spaces. We propose an equivalent axiomatic definition, which is shorter, more practical, and closer to the axiom of Bruhat–Tits buildings. Our main tool to prove the equivalence of the axioms is the study of the convexity properties in masures.

A Tits polygon is a bipartite graph in which the neighborhood of every vertex is endowed with an “opposition relation” satisfying certain properties. Moufang polygons are precisely the Tits polygons in which these opposition relations are all trivial. There is a standard construction that produces a Tits polygon whose opposition relations are not all trivial from an arbitrary pair $(\unicode[STIX]{x1D6E5},T)$, where $\unicode[STIX]{x1D6E5}$ is a building of type $\unicode[STIX]{x1D6F1}$, $\unicode[STIX]{x1D6F1}$ is a spherical, irreducible Coxeter diagram of rank at least $3$, and $T$ is a Tits index of absolute type $\unicode[STIX]{x1D6F1}$ and relative rank $2$. A Tits polygon is called $k$-plump if its opposition relations satisfy a mild condition that is satisfied by all Tits triangles coming from a pair $(\unicode[STIX]{x1D6E5},T)$ such that every panel of $\unicode[STIX]{x1D6E5}$ has at least $k+1$ chambers. We show that a $5$-plump Tits triangle is parametrized and uniquely determined by a ring $R$ that is alternative and of stable rank $2$. We use the connection between Tits triangles and the theory of Veldkamp planes as developed by Veldkamp and Faulkner to show existence.

Let $T$ be a locally finite tree without vertices of degree $1$. We show that among the closed subgroups of $\text{Aut}(T)$ acting with a bounded number of orbits, the Chabauty-closure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of $T$ have degree ${\geqslant}3$, then the set of isomorphism classes of topologically simple closed subgroups of $\text{Aut}(T)$ acting doubly transitively on $\unicode[STIX]{x2202}T$ carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid in the context of automorphism groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented.

Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$. We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$, the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits.

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 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.

We prove simplicity for incomplete rank 2 Kac—Moody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs. We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite). Nevertheless we use the fact that the latter groups are just infinite (modulo center).

In this paper, we establish that complete Kac–Moody groups over finite fields are abstractly simple. The proof makes essential use of Mathieu and Rousseau’s construction of complete Kac–Moody groups over fields. This construction has the advantage that both real and imaginary root spaces of the Lie algebra lift to root subgroups over arbitrary fields. A key point in our proof is the fact, of independent interest, that both real and imaginary root subgroups are contracted by conjugation of positive powers of suitable Weyl group elements.

We give an explicit construction of the Ree groups of type G2 as groups acting on mixed Moufang hexagons together with detailed proofs of the basic properties of these groups contained in the two fundamental papers of Tits on this subject (see [7] and [8]). We also give a short proof that the norm of a Ree group is anisotropic.

Given a complete $\text{CAT}(0)$ space $X$ endowed with a geometric action of a group $\Gamma $, it is known that if $\Gamma $ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We prove the converse of this statement in the special case where $X$ is a convex subcomplex of the $\text{CAT}(0)$ realization of a Coxeter group $W$, and $\Gamma $ is a subgroup of $W$. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the $\text{CAT}(0)$ realization of arbitrary Tits buildings.

For cohomological (respectively homological) coefficient systems ${\mathcal F}$ (respectively ${\mathcal V}$) on affine buildings X with Coxeter data of type $\widetilde{A}_d$, we give for any $k\ge1$ a sufficient local criterion which implies $H^k(X,{\mathcal F})=0$ (respectively $H_k(X,{\mathcal V})=0$). Using this criterion we prove a conjecture of de Shalit on the acyclicity of coefficient systems attached to hyperplane arrangements on the Bruhat–Tits building of the general linear group over a local field. We also generalize an acyclicity theorem of Schneider and Stuhler on coefficient systems attached to representations.

The spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynimial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynimials.