We introduce a new method to calculate local normal zeta functions of finitely generated, torsion-free nilpotent groups. It is based on an enumeration of vertices in the Bruhat-Tits building for Sl_n(Q_p). It enables us to give explicit computations for groups of class 2 with small centres and to derive local functional equations. Examples include formulae for non-uniform normal zeta functions.
Consider matrices of order $k+N$ over $p$-adic field determined up to conjugations by elements of $GL$ over $p$-adic integers. We define a product of such conjugacy classes and construct the analog of characteristic functions (transfer functions), they are maps from Bruhat-Tits trees to Bruhat-Tits buildings. We also examine categorical quotient for usual operator colligations.
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.
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…
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.
Let GF denote the rational points of a semisimple group G over a non-archimedean local field F, with Bruhat-Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of GF in the Chabauty topology, which enables to compactify the vertices of X. We obtain a structure theorem showing that the Bruhat-Tits buildings of the Levi factors all lie in the boundary of the compactification. Then we obtain an identification theorem with the polyhedral compactification (previously defined in analogy with the case of symmetric spaces). We fin…
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.
We construct a compactification of the Bruhat-Tits building associated to the group PGL(V) which can be identified with the space of homothety classes of seminorms on V endowed with the topology of pointwise convergence. Then we define a continuous map from the projective space to this compactification which extends the reduction map from Drinfeld's p-adic symmetric domain to the building.
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.…
Based on Broise-Alamichel and Paulin's work on the Gauss map corresponding to the principal convergents, we continue the study of the Gauss map via Farey maps to contain all the intermediate convergents. We define the geometric Farey map, which is given by time-1 map of the geodesic flow. We also define algebraic Farey maps, better suited for arithmetic properties, which produce all the intermediate convergents. Then we obtain the ergodic invariant measures for the Farey maps and the convergent speed.
Let $K$ be an algebraic function field of one variable with constant field $k$ and let $C$ be the Dedekind domain consisting of all those elements of $K$ which are integral outside a fixed place $\infty$ of $K$. When $k$ is finite the group $GL_2(C)$ plays a central role in the theory of Drinfeld modular curves analagous to that played by $SL_2(Z)$ in the classical theory of modular forms. When $k$ is finite (resp. infinite) we refer to a group $GL_2(C)$ as an arithmetic (resp. non-arithmetic) Drinfeld modular group. Associated with $GL_2(C)$ is its Bruhat-Tits tree, $T$. The structure of the …
We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If G is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of G and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of G. We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation and that it is related to the tropicalization of the hypersurface given by the character …
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.
Let K be a function field with constant field k and let "infinity" be a fixed place of K. Let C be the Dedekind domain consisting of all those elements of K which are integral outside "infinity". The group G=GL_2(C) is important for a number of reasons. For example, when k is finite, it plays a central role in the theory of Drinfeld modular curves. Many properties follow from the action of G on its associated Bruhat-Tits tree, T. Classical Bass-Serre theory shows how a presentation for G can be derived from the structure of the quotient graph (or fundamental domain) G\T. The shape of this quot…
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…
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…
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…
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 $k$ denote a field with nontrivial discrete valuation. We assume that $k$ is complete with perfect residue field. Let $G$ be the group of $k$-rational points of a reductive, linear algebraic group defined over $k$. Let $\gg$ denote the Lie algebra of $G$. Fix $r \in \R$. Subject to some restrictions, we show that the set of distinguished degenerate Moy-Prasad cosets of depth $r$ (up to an equivalence relation) parametrizes the nilpotent orbits in $\gg$.
In our previous paper "Bruhat-Tits theory from Berkovich's point of view. I ? Realizations and compactifications of buildings", we investigated realizations of the Bruhat-Tits building B(G,k) of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry, and we studied in detail the compactifications of the building which arise from this point of view. In this paper, we give a representation theoretic flavor to these compactifications, following Satake's original constructions for Riemannian symmetric spac…
We call a non-discrete Euclidean building a Bruhat-Tits space if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root datum of the building at infinity. This is the class of non-discrete Euclidean buildings introduced and studied by Bruhat and Tits. We give the complete classification of Bruhat-Tits spaces whose building at infinity is the fixed point set of a polarity of an ambient building of type B_2, F_4 or G_2 associated with a Ree or Suzuki group endowed with the usual root datum. (In the B_2 and G_2 cases, this fixed point set i…
Let $k$ be a field with a nontrivial discrete valuation which is complete and has perfect residue field. Let $G$ be the group of $k$-rational points of a reductive, linear algebraic group $\textbf{G}$ equipped with an involution $θ$ defined over $k.$ Let $\mathfrak{p}$ denote the $(-1)$-eigenspace in the decomposition of the Lie algebra of $G$ under the differential $dθ.$ If $\textbf{H}$ is a subgroup of $\textbf{G}^θ$, the set of $θ$-fixed points, which contains the connected component of $\textbf{G}^θ,$ then $H=\textbf{H}(k)$ acts on $\mathfrak{p}$, which we treat as a symmetric space. Let $…
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…
This notes are additional remarks to an article of Broussous and Stevens [arXiv:math/0402228v1]. We consider a unitary group G over a non-Archimedean local field k_0 of residue characteristic different from two and an element β of the Lie algebra \mf{g} of G. Let H be the centralizer of β in G. We further assume k_0[β] to be semisimple. We prove that there is an affine H-equivariant map between the Bruhat-Tits buildings B(H)\ra B(G) which is compatible with the Lie-algebra filtrations (CLF) and maps apartments into apartments. The map is toral if β is separable. For simplicity let us now assum…
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…
In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schutzenberger and Reading.
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）
P. Broussous and S. Stevens studied maps between enlarged Bruhat-Tits buildings to construct types for p-adic unitary groups. They needed maps which respect the Moy-Prasad filtrations. That property is called (CLF), i.e. compatibility with the Lie algebra filtrations. In the first part of this thesis we generalise their results on such maps to the Quaternion-algebra case. Let k0 be a p-adic field of residue characteristic not two. We consider a semisimple k0-rational Lie algebra element beta of a unitary group G:=U(h) defined over k0 with a signed hermitian form h. Let H be the centraliser of …
This article answers a question that naturally arises from the articles by Grabitz and Broussous "Pure elements and intertwining classes of simple strata in local central simple algebras" and Broussous and Lemaire "Buildings of GL(m,D) and Centralizers". For an Azumaya-Algebra A over a non-Archimedean local field F, Grabitz and Broussous have introduced embedding invariants for field embeddings, that is for pairs (E,a), where E is a field extension of F in A, and $a$ is a hereditary order which is normalised by E^x. On the other hand if we take such a field extension E and define B to be the c…
We study the action on the Bruhat-Tits tree of unit groups of maximal orders in certain quaternion algebras over $\mathbb{F}_q(T)$ and discuss applications to arithmetic geometry and group theory.
In this note we compute the fundamental domain of the action on the corresponding Bruhat-Tits tree of the arithmetic subgroup of PGL2 with respect to an arbitrary place of the rational function field.
Let $F$ be a discretely valued complete field with valuation ring $\mathcal{O}_F$ and perfect residue field $k$ of cohomological dimension $\leq 1$. In this paper, we generalize the Bruhat decomposition in Bruhat and Tits from the case of simply connected $F$-groups to the case of arbitrary connected reductive $F$-groups. If $k$ is algebraically closed, Haines and Rapoport define the Iwahori-Weyl group, and use it to solve this problem. Here we define the Iwahori-Weyl group in general, and relate our definition of the Iwahori-Weyl group to that of Haines and Rapoport. Furthermore, we study the…
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…
For a split Kac-Moody group (in J. Tits' definition) over a field endowed with a real valuation, we build an ordered affine hovel on which the group acts. This construction generalizes the one already done by S. Gaussent and the author when the residue field contains the complex field [Annales Fourier, 58 (2008), 2605-2657] and the one by F. Bruhat and J. Tits when the group is reductive. We prove that this hovel has all properties of ordered affine hovels (masures affines ordonnées) as defined in [Rousseau, ArXiv 0810.4241]. We use the maximal Kac-Moody group as defined by O. Mathieu and we p…
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.
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.
