Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.
This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry canbe used to compactify buildings. We discuss in detail the example of the special linear group. Moreover, we give an intrinsic description of Bruhat-Tits buildings in the framework of non-Archimedean analytic geometry.
Given an adjoint semisimple group $G$ over a local field $k$, we prove that the maximal Satake-Berkovich compactification of the Bruhat-Tits building of $G$ can be identified with the one obtained by embedding the building into the Berkovich analytification of the wonderful compactification of $G$, extending previous results of Rémy, Thuillier and Werner. In the process, we use the characterisation of the wonderful compactification in terms of Hilbert schemes given by Brion to extend the definition of the wonderful compactification to the case of a non-necessarily split adjoint semisimple grou…
Given a semisimple group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat-Tits builidng into the analytic space associated to the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.
Given a semisimple group over a local field of residual characteristic p, its topological group of rational points admits maximal pro-p-subgroups. Quasi-split simply-connected semisimple groups can be described in the combinatorial terms of valued root groups, thanks to Bruhat-Tits theory. In this context, it becomes possible to compute explicitly a minimal generating set of the (all conjugated) maximal pro-p-subgroups thanks to parametrizations of a suitable maximal torus and of corresponding root groups. We show that the minimal number of generators is then linear with respect to the rank of…
We extend (scheme-theoretic) Bruhat-Tits theory to quasi-reductive groups i.e. with trivial split unipotent radical over discretely valued henselian non-archimedean fields $K$, whose ring of integers is excellent and residue field is perfect, building on Bruhat-Tits I et II and the classification of pseudo-reductive groups by Conrad-Gabber-Prasad. The buildings had been constructed previously by Solleveld in a different manner, but they ultimately coincide with ours. We were motivated by the geometry of affine Grassmannians and give some applications at the end (compare also with previously su…
In a Bruhat-Tits building of split classical type (that is, of type $A_n$, $B_n$, $C_n$, $D_n$, and any combination of them) over a local field, the simplicial volume counts the vertices within the given simplicial distance from a special vertex. This paper aims to study the asymptotic growth of the simplicial volume. A formula of the simplicial volume is deduced from the theory of concave functions. Then the dominant term in its asymptotic growth is found using the theory of $q$-exponential polynomials developed in this paper.
Given a quasi-reductive group $G$ over a local field $k$, using Berkovich geometry, we exhibit a family of $G(k)$-equivariant compactifications of the Bruhat-Tits building $\mathcal B(G, k)$, constructed and investigated by Solleveld and Lourenço. The compactification procedure consists in mapping the building in the analytification $G^{\mathrm{an}}$ of $G$, then composing this map with the projections from $G^{\mathrm{an}}$ to its (in general non-compact) pseudo-flag varieties $(G/P)^{\mathrm{an}}$, for $P$ ranging among the pseudo-parabolic subgroups of $G$. This generalises previous constru…
A vertex-transitive graph X is called local-to-global rigid if there exists R such that every other graph whose balls of radius R are isometric to the balls of radius R in X is covered by X. Let $d\geq 4$. We show that the 1-skeleton of an affine Bruhat-Tits building of type $\widetilde A_{d-1}$ is local-to-global rigid if and only if the underlying field has characteristic 0. For example the Bruhat-Tits building of $SL(d,F_p((t)))$ is not local-to-global rigid, while the Bruhat-Tits building of $SL(d,Q_p)$ is local-to-global rigid.
Let $Δ$ be a locally finite thick building of type $\tilde{A}_2$. We show that, if the type-preserving automorphism group $\mathrm{Aut}(Δ)^+$ of $Δ$ is transitive on panels of each type, then either $Δ$ is Bruhat--Tits or $\mathrm{Aut}(Δ)$ is discrete. For $\tilde{A}_2$-buildings which are not panel-transitive but only vertex-transitive, we give additional conditions under which the same conclusion holds. We also find a local condition under which an $\tilde{A}_2$-building is ensured to be exotic (i.e.\ not Bruhat--Tits). It can be used to show that the number of exotic $\tilde{A}_2$-buildings…
We study the distance on the Bruhat-Tits building of the group $\mathrm{SL}_d(\mathbb{Q}_p)$ (and its other combinatorial properties). Coding its vertices by certain matrix representatives, we introduce a way how to build formulas with combinatorial meanings. In Theorem 1, we give an explicit formula for the graph distance $δ(α,β)$ of two vertices $α$ and $β$ (without having to specify their common apartment).Our main result, Theorem 2, then extends the distance formula to a formula for the smallest total distance of a vertex from a given finite set of vertices. In the appendix we consider the…
Let $K$ be the function field of a curve $C$ over a field $\mathbb{F}$ of either odd or zero characteristic. Following the work by Serre and Mason on $\mathrm{SL}_2$, we study the action of arithmetic subgroups of $\mathrm{SU}(3)$ on its corresponding Bruhat-Tits tree associated to a suitable completion of $K$. More precisely, we prove that the quotient graph "looks like a spider", in the sense that it is the union of a set of cuspidal rays (the "legs"), parametrized by an explicit Picard group, that are attached to a connected graph (the "body"). We use this description in order to describe t…
Consider an infinite homogeneous tree $T_n$ of valence $n+1$, its group $Aut(T_n)$ of automorphisms, and the group $Hie(T_n)$ of its spheromorphisms (hierarchomorphisms), i.~e., the group of homeomorphisms of the boundary of $T_n$ that locally coincide with transformations defined by automorphisms. We show that the subgroup $Aut(T_n)$ is spherical in $Hie(T_n)$, i.~e., any irreducible unitary representation of $Hie(T_n)$ contains at most one $Aut(T_n)$-fixed vector. We present a combinatorial description of the space of double cosets of $Hie(T_n)$ with respect to $Aut(T_n)$ and construct a 'ne…
Let C be a smooth, projective and geometrically integral curve defined over a finite field F. For each closed point P of C, let R be the ring of functions that are regular outside P, and let K be the completion at P of the function field of C. In order to study groups of the form GL2(R), Serre describes the quotient graph GL2(R)\t, where t is the Bruhat-Tits tree defined from SL2(K). In particular, Serre shows that GL2(R)\t is the union of a finite graph and a finite number of ray shaped subgraphs, which are called cusps. It is not hard to see that finite index subgroups inherit this property.…
Non-positively curved spaces admitting a cocompact isometric action of an amenable group are investigated. A classification is established under the assumption that there is no global fixed point at infinity under the full isometry group. The visual boundary is then a spherical building. When the ambient space is geodesically complete, it must be a product of flats, symmetric spaces, biregular trees and Bruhat--Tits buildings. We provide moreover a sufficient condition for a spherical building arising as the visual boundary of a proper CAT(0) space to be Moufang, and deduce that an irreducible…
A quotient of the Bruhat-Tits building classifying maximal matrix orders over curves
Universidad de Chile, Facultad de Ciencias（智利大学理学院）；Universidad de Talca, Instituto de Matemáticas（塔尔卡大学数学研究所）；Facultad de Ingeniería y Ciencias, Universidad Diego Portales（迭戈·波特莱斯大学工程与理学院）
The purpose of this paper is to link anisotropy properties of an algebraic group together with compactness issues in the topological group of its rational points. We nd equivalent conditions on a smooth ane algebraic group scheme over a non-Archimedean local eld for the associated rational points to admit maximal compact subgroups. We use the structure theory of pseudo-reductive groups provided, whatever the characteristic, by Conrad, Gabber and Prasad. We also investigate thoroughly maximal prop subgroups in the semisimple case, using Bruhat-Tits theory.
This paper provides a unified combinatorial framework to study orbits in certain affine flag varieties via the associated Bruhat-Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine We…
We introduce structures which model quotients of buildings by type-preserving group actions. These structures, which we call W-groupoids for W a Coxeter group, generalize Bruhat decompositions, chambers systems of type M, Tits amalgams, and buildings themselves. We define the fundamental group of a W-groupoid, and characterize buildings as connected simply connected W-groupoids. We give a brief outline of covering theory of W-groupoids, which produces buildings as universal covers equipped with an action of the fundamental group. The local-to-global theorem of Tits concerning spherical 3-resid…
Given a locally finite leafless tree $ T $, various algebraic groups over local fields might appear as closed subgroups of $ \text{Aut} (T)$. We show that the set of closed cocompact subgroups of $ \text{Aut} (T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $ \text{Aut} (T)$. This is done via a study of the integral Bruhat$-$Tits model of $ \text{SL}_2 $ and $ \text{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…
In this paper, we prove some finiteness results about split Kac-Moody groups over local non-archimedean fields. Our results generalize those of "An affine Gindikin-Karpelevich formula" by Alexander Braverman, Howard Garland, David Kazhdan and Manish Patnaik. We do not require our groups to be affine. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group.
