Bc. Dominik Lachman. Bruhat-Tits buildings

Transcription

1 MASTER THESIS Bc. Dominik Lachman Bruhat-Tits buildings Department of Algebra Superisor of the master thesis: Study programme: Study branch: Mgr. Vítězsla Kala, Ph.D. Mathematics Mathematical structure Prague 2017

2 I declare that I carried out this master thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that the Charles Uniersity has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 subsection 1 of the Copyright Act. In... date... signature of the author i

3 Title: Bruhat-Tits buildings Author: Bc. Dominik Lachman Department: Department of Algebra Superisor: Mgr. Vítězsla Kala, Ph.D., Department of Algebra Abstract: Bruhat-Tits buildings are a fundamental concept in the study of linear algebraic groups oer general fields. The general goal of this thesis is to introduce buildings in the basic case of SL d (Q p ) and to explicitly describe some of their geometrical and combinatorial properties building are abstract simplicial complexes. After the general construction (Chapter 1) we focus in detail to the case of SL 2 (Q p ). We work with simplices using certain matrix representaties. We explicitly describe the building and gie a formula for graph distance. In Chapter 3 we consider the general case SL d (Q p ), d 2. There we introduce a new concept of distance formulas. In Chapter 4 we proe some theorems which are satisfied by buildings in general. Chapter 5 studies the problem of determining so-called gallery distance of two simplices. In the last Chapter 6 we generalize the distance formulas to the case of three ertices. Keywords: Bruhat-Tits buildings, group SL d (Q p ), graph distance in apartments ii

4 I would like to thank my superior, Vítězsla Kala, for his adices and the time he spent reading this thesis. But also for his friendliness in teaching me and help in my study beyond this thesis. iii

5 Contents List of Notations 2 Introduction 4 1 Construction of the building for group SL d (Q p ) Algebraic background Geometric-combinatorial background General definition of building Construction of building for group SL d (Q p ) Matrix representaties The case of SL 2 (Q p ) Parametrization of apartments passing through the origin ertex α Description of the graph structure of the building B(GL 2 (Q p )) Some formulas Distance formulas Localized form and distance-formulas The relatie position of two ertices Some combinatorial and geometrical properties of apartments in B(SL d (K)) Colouring and elementary path Intersection of two apartments and canonical retraction on apartments Intersection of two apartments Applications Gallery distance Matrix representaties of chambers of a gien apartment Gallery distance of two ertices Gallery distance of two chambers Distance formulas for three ertices Some distances on the set of ertices in B Length of minimal three connecting three ertices Bibliography 83 1

6 List of Notations Z N N 0 Q R Q p Z p R S d GL d (K) field of integers natural numbers natural numbers and zero field of rational numbers field real numbers field of p-adic numbers ring of p-adic integers group of all inertible elements in ring R Symmetric group on a set of d elements group of regular d d matrices oer field K SL d (K) matrices from GL d (K) haing determinant equal to 1 GP d (K) Diag d (K) E I A i,j A i, A,j Row R (A) set of generalized permutation matrices oer field K set of diagonal matrices oer field K the canonical basis of ector space K d identity matrix entries of matrix A the i-th row of matrix A the j-th column of matrix A R-lattice generated by rows of matrix A B(SL d (Q p )) building for group GL d (Q p ) ν additie p-adic aluation Notions introduced in this thesis: A A a A c A, A B, B A p m A:i,B:d i λ i,β:α δ(α, β) δ el (α, β) ertex represented by matrix A apartment represented by matrix A chamber represented by matrix A left and right wing of apartment A left and right wing of the building B aluation of det(a) Definition 3.1.1, also referred as ma:i B:d i or m d i i the i-th relatie coordinate of ertex β to the α (Definition 3.2.9) graph distance of ertices α and β elementary graph distance of ertices α and β Ω(C 1, C 2 ) the gallery-distance of chambers C 1 and C 2 (Def ) 2

7 Through the whole thesis we follow this conention in using symbols: A, B, C,... matrix representaties of ertices, apartments, chambers α, β, γ,... ertices B building A apartment C chamber S simplex 3

8 Introduction Bruhat-Tits buildings are a fundamental concept in the study of linear algebraic groups (i.e., groups of matrices) oer general fields, giing tool to describe their representations and certain subgroups. The general goal of this thesis is to introduce buildings in the basic case of SL d (Q p ) and to explicitly describe some of their geometrical and combinatorial properties. Buildings were introduced by Tits to inestigate exceptional groups oer any field from a geometric point of iew. He described the concept in seeral papers published during 1950s, where he do not yet use the terminology of buildings using present-day. His effort in studying group geometrically could be considered as continuing in Klein s Erlangen program. Tits construction associates to each semi-simple algebraic group oer any field the spherical building. It is a certain simplicial complex on which the group acts by simplicial automorphisms. Maximal parabolic subgroups are in one-to-one correspondence with maximal simplices, called chambers, and similarly maximal splitting tori correspond to subcomplexes called apartments. Apartments are all isomorphic and they are geometrically spheres chopped into simplices (hence the name spherical building). Later Tits together with Bruhat introduced so called Euclidean buildings, following the work of Iwahori and Matsumoto inestigating Lie groups oer local non-archimedian fields. Euclidean buildings are useful to study reductie algebraic groups oer local non-archimedian field in a way analogue to the the spherical case. Apartments in this case are Euclidean spaces chopped into simplices (e.g., triangulations of R n ). One can also treat buildings as an independent notion, that is, simplicial complexes satisfying seeral axioms. It is interesting that eery building of dimension greater or equal to 2 arises as a building associated to group by Tits construction. And such group is essentially determined by the building. This holds for both cases: spherical buildings and semi-simple algebraic groups, and Euclidean buildings and algebraic reductie groups oer local non-archimedian field. By saying that buildings reflect the structure of groups geometrically, we mean the incidence geometry. For example two parabolic subgroups hae non-empty intersection if the corresponding chambers intersect in a simplex. In this thesis we consider (Euclidean) buildings B(SL d (Q p )) associated to groups SL d (Q p ), d 2. Our aim is to describe the buildings explicitly and inestigate some of their combinatorial properties like graph distance (in its 2- dimensional substructure). After the general construction (Chapter 1) we will focus in detail to the easiest case SL 2 (Q p ). We will work in coordinates depending on the choice of basis: we will work with simplices using so called matrix representaties. This approach may be less elegant than a more algebraic, coordinate-free one, since we will hae to concern ourself with the ambiguities in matrix representaties. Howeer, at the end of the chapter we introduce two formulas, whose symmetry absorbs all the ambiguity of coordinate-like data and hence hae geometric meanings. One formula measures graph distance of two ertices, the second one counts the number of edges shared by two apartments. In Chapter 3 we success in generalizing the formula of graph distance in seeral 4

9 ways for all d 2, in the so called distance formulas. Although the distance formulas are not computation-friendly (similarly as the well-known formula for matrix inersion using cofactors), they proides seeral benefits for the theory, for example we gie in Theorem an explicit construction of canonical retraction to apartments (i.e., retraction of the whole building to a gien apartment) which exists for any building. In Chapter 4 we describe the intersection of two apartments represented by certain matrices, by inequalities (Proposition 4.1.3). Then we use this description to proe some properties which buildings satisfy in general (e.g., that eery two simplices hae some apartment in common and the existence of canonical retraction to apartments, already mentioned). The Chapter 5 studies the problem determining the length of minimal gallery containing gien two ertices or chambers. It takes some technical effort to wellestablish matrix representaties of chambers well, but the idea standing behind proof is purely combinatorial. In the last Chapter 6 we generalize the formula for graph distance to a lower bound on the length of minimal tree connecting three ertices. Finally, we further generalize it to certain formula concerning any number of ertices. 5

10 1. Construction of the building for group SL d (Q p ) In preparation of this thesis we hae used mostly [1]. An exception is the general construction of buildings (Section 3), which follows [2]. 1.1 Algebraic background Through the whole thesis we assume the knowledge of basic facts about p-adic numbers, which we will use without reference. We gie quick oeriew (without proofs) to p-adic numbers in this section. Let p Z be a prime number. On Z define aluation ν p by ν p (a) = max{k N 0 a (p) k }, (1.1) where (p) is the prime ideal of Z generated by p. And further extend ν p to the whole Q by ν p ( a b ) = ν p(a) ν p (b), for each a, b Z p. On Q define norm p : Q R 0, gien by prescription a p = p νp(a) for each a Q (i.e., hight powers of p are small). Finally, define the field of p-adic numbers Q p as the completion of Q with respect to norm p. We can think of Q p as the infinite power series: Each a Q p can be uniquely written in form a = a i p i, (1.2) i=n where n Z and a i s are taken from {0,..., p 1} and a n 0. The aluation ν p continuously (with respect to topology induced by norm p ) extends to the whole Q p by prescription (1.1) (e.g., gien a from expression (1.2), ν p (a) equals n). Next, define the ring of p-adic integers Z p as the completion of Z in Q p. See that Z p = {a Q p ν p (a) 0}. The inertible elements in Z p are exactly those haing zero aluation Z p = {a Q p ν p (a) = 0}. An important inequality called triangle inequality is this: For eery a 1,..., a n Q p we hae ν p (a a n ) min{ν p (a 1 ),..., ν p (a n )}. (1.3) Next define others basic algebraic concepts, used in this thesis: By Z p -lattice we mean any finitely generated Z p -submodule of some Q p -ector space. A key 6

11 fact is that eery Z p -lattice L has Z p -basis. This is consequence of well-known Theorem of elementary diisors [3]. Let R be ring, then by GL d (R) we denote the group of all inertible (d d)- matrices oer R. For example, GL d (Z p ) are those matrices haing determinant in Z p. By SL d (R) we denote all (d d)-matrices oer ring R haing determinant equal to 1. We will also use group of diagonal (d d)-matrices denoted by Diag d (R). 1.2 Geometric-combinatorial background Gien any (possibly infinite) set S, by abstract simplicial complex we mean collection of non-empty finite subsets of S, such that: wheneer g f then g. The dimension of f is by definition f 1 and the elements of are called faces. The 0-dimension faces correspond to elements of S and we call them ertices. The 1-dimensional faces are called edges. By subcomplex of abstract simplicial complex we mean any simplicial complex such that. Gien any face f, define subcomplex S of as a collection S = {g g f}. Of course f is face of S. We call such subcomlexes simplices. Clearly each simplex is uniquely gien by its ertices. See, that any abstract simplicial complex is union of its simplices. Gien any simplicial complex, the subcomplex consisting of all two-dimensional faces is essentially graph (we will treated it as a graph). One of our aim is to inestigate a graph sub-structure of buildings. We will need following notion: Gien graph G = (V, E), by walk in G we mean any sequence 0, e 1, 1,..., n, where i s are ertices (i.e., elements of V ) and e i s are edges (i.e., elements of E). And a path in graph G is a walk without repetition in its ertices. 1.3 General definition of building Building is a simplicial complex satisfying seeral conditions forcing structural symmetry. Defining buildings let us introduce the following notions which are standard in the theory: Definition For simplicial complex of dimension d < we call any simplex of dimension d a chamber. And we say that two chambers are adjacent if they intersect in a (d 1)-dimensional simplex. Note that adjacent chambers differ by exactly 1 ertex. Definition A sequence ρ 0,..., ρ n of chambers is called a gallery if for each 0 i < n, the chambers ρ i and ρ i+1 are adjacent. The gallery is minimal if there is no shorter gallery containing ρ 0 and ρ n. 7

12 Here is the definition of (thick) building: Definition A simplicial complex B is a building of dimension d if there is a family A of sub-complexes called apartments such that: (Ax. i) for each apartment A A and (d 1)-dimensional simplex S belonging to A there are exactly two chambers in A that contain S, (Ax. ii) in eery apartment A A, each pair of chambers is connected by a gallery, (Ax. iii) each pair of simplices has some common apartment containing both of them, (Ax. i) wheneer two apartments A and A share two simplices S and S, there is simplicial isomorphism which maps A onto A, and which fixes both S and S pointwise (that is, also the ertices of both simplices). Furthermore, a building is thick if following holds (Ax. i) eery k-simplex, k < d, is contained in at least three chambers Some immediate consequences: (iii) implies each simplex lies in some apartments, hence a building is union of its apartments. Axiom (i) together with (iii) implies that eery maximal simplex is of dimension d. By axiom (iii) for eery pair of apartments there is an apartment haing non-empty intersection with both of them and so all three apartments are isomorphic by (i) (i.e., all apartments are isomorphic). For a fixed simplex S, the whole building is the union of all apartments containing S, as asserted by (iii). Notice that if we leae (i), a building which consists of only one it s apartment satisfy all axioms. Howeer, if we require (i), the collection of apartments is rich, since the apartments branch out in each simplex. In the Figure 1.1, there is initial part of regular infinite tree of degree 3. This tree is building where we take for apartments all the infinite paths without ends (note that chambers are in this case edges). Eery building gies rise to group of certain simplicial automorphisms stabilising the family of apartments. The conerse direction holds, besides others, for reductie algebraic groups oer local non-archimedean field, we call such buildings Bruhat-Tits buildings. In this thesis we shall study the buildings associated to groups SL d (Q p ). 1.4 Construction of building for group SL d (Q p ) In this section we shall construction building B = B(SL d (Q d )), where Q p is the field of p-adic rational numbers. The construction is non-triial and some condition from Definition are not clear to hold at all. In the thesis, we shall 8

13 Figure 1.1: Example of building. work with the building as explicitly constructed in this section and will not use the preious axiomatic definition. Howeer, we will, besides other things, proe all the axioms: (i) in Theorem 3.3.3, (ii) follows from Proposition where we gie a formula for length of minimal galleries, (iii) in Theorem 4.2.4, (i) in Corollary and (i)(thickness) in Theorem We will construct B as a topological simplicial complex, but we will not erify it really is a topological simplicial complex in detail, since the construction is standard [2]. Then we concentrate on B as an abstract simplicial complex, because this iew better suits to our aim: inestigating combinatorial properties of B. Through whole thesis Z p denote p-adic integers and ν p additie p-adic aluation, usually referred only by ν. Further for purpose of this chapter denote by V the ector space Q d p, where d 2 is integer. We will consider V as a ector space of row ectors. Definition Call α : V R { } (additie) norm if for each x, y V and λ Q p following conditions hold: (i) α(x + y) min{α(x), α(y)}, (ii) α(λx) = ν(λ) + α(x), (iii) α(x) = x = 0. Denote by N (V ) the set of all norms on V. Remark In fact N can be iewed as building for GL d (Q p ), but it does not has simplicial structure, only poly-simplicial. We shall not use this fact. After while we define building of SL d (Q p ) as a quotient of N. Lemma ([4],Theorem III.2.6). For each norm α N there is basis 1,..., d 9

14 of V and real numbers r 1,..., r d, such that α(x x d d ) = min{ν(x 1 ) + r 1,..., ν(x d ) + r d }, (1.4) for all x 1,..., x d Q p. Conersely, for each choice of d-tuple of real numbers r 1,..., r d and basis 1,..., d formula (1.4) defines (additie) norm. We shall call such basis splitting basis for norm α. We are especially interested in so called special norms. Definition Norm α N is called special if there is Z p -lattice L such that α(x) = max{k Z x p k L}. (1.5) Following lemma ensures correctness of preious definition and describes all special norms. Lemma Let 1,..., d be basis of V = Q d p, and L be the Z p -lattice generated by i s. Then norm α defined by equation (1.4) with r 1 = = r d = 0 and L satisfy equality (1.5). Conersely, eery Z p -lattice L V defines a norm. Proof. Assume that = x x d d and that α is defined by (1.4), we hae α() = min i {ν(x i )}. Hence if α() = k, ector belongs to Z p p k 1,..., p k d = p k L. For each integer l > k ector does not belong to p l L because this Z p -lattice has ectors p l i s as Z p -basis with respect to which has coefficients p l x i s and these are not all in Z p. By well-known Theorem of elementary diisors [3] are all torsion-free modules oer discrete aluation rings free. Hence, each Z p -lattice L V has Z p -basis, which defines by equation (1.4) norm α L. According to first part α L satisfies equation (1.5). Haing any Z p -lattice L we can use formula (1.5) to define additie norm α L. According to Lemma eery Z p -basis of L is splitting basis for α L. Obseration Special norms are exactly those haing as image Z. Norm gien by formula (1.4) is special if and only if all r i s are integers. Proof. Wheneer norm α is gien by (1.4) with some r i not integer, clearly α( i ) = r i. On the other hand if all r i s are integers, we can define i = p r i i for all i s, and compute: α(x x d d) = α(x 1 p r x d p r d d ) = = min{ν(p r 1 x 1 ) + r 1,..., ν(p r d x d ) + r d } = min{ν(x 1 ),..., ν(x d )}. Hence wheneer r i s are integers, corresponding norm is special by Lemma and its image is Z. Obseration Formula (1.5) defines one-to-one correspondence between special norms and Z p -lattices. Proof. We hae to erify that two different lattices gies efferent norms. If Z p - lattices L, K are such that there is L \ K and α, β are corresponding special norms, then α() 0 > β(). Hence different Z p -lattices gies different norms. 10

15 On N (V ) define relation of equialence: α β if there is r R such that α(x) = β(x) + r for each x V. Finally define B = B(SL d (Q p )) := N (V )/. Obseration Let α L and α K be special norms corresponding to lattices L and K. Then α L α K if and only if there is integer i such that L = p i K. Proof. Concerning special norms, on side of lattices (see Obseration 1.4.7) the reduction identifies following lattices for each lattice L. p 1 L L pl, Definition We call B the building for group SL d (Q p ), more precisely denoted by B(SL d (Q p )). Equipped with following simplicial structure: For ertices take all reduction classes of special norms and for each (k +1)-tuple of Z p -lattices L 0 L 1 L k pl 0 (1.6) add k-simplex, whose ertices are that gien by Z p -lattices L i s. Lemma Eery maximal simplex of B(SL d (Q p )) is of dimension d 1. Specially the whole building has dimension d 1. Proof. By well know correspondence theorem [5](Proposition 2.9.) for modules we hae one-to-one correspondence between Z p -lattices K, L K pl and non-triial Z p -submodules of L/pL, presering inclusion. We can consider the later as F p = Z p /pz p -ector space of dimension d. Eery chain (1.6) gies filtration of L/pL of length k + 1. Clearly, maximal filtrations of ector space L/pL has length d (the dimension of L/pL oer F p ). And eery filtration of ector space can be completed into maximal one, that corresponds to fact that eery chain (1.6) can be completed into chain with k = d 1. (The apartment of course depends on the choice of basis 1,..., d.) Definition Let 1,..., d be basis of V. For each n-tuple of real numbers r = (r 1,..., r d ), define additie norm α r by formula (1.4). Call the image of composition of mappings R d N B d sending r [α r ] an apartment. Recalling Obseration following is clear: Obseration The ertices in an apartment gien by basis 1,..., d are those gien by Z p -lattices Z p x 1 1,..., x d d, for x 1,..., x d Q p. It is clearly enough to take for x i s powers of p. Now we can interpret Lemmas as: Eery element of B lies in some apartment so the building is a union of its apartments. Definition The action of GL d (Q p ) on B is a group homomorphism ρ : GL d (Q p ) Aut(B), by Aut(B) we mean group of simplicial automorphisms, defined as follows: For a norm α and g GL d (Q p ), let g α be the norm defined by (g α)() = α( g), where is iewed as a row ector and g is matrix multiplication. For [α] B we then define ρ g ([α]) = [g α]. 11

16 Remark The action is well defined: Verification that g α is norm for each g GL d (Q p ) and α N (V ) is straightforward. So is the fact that ρ is group homomorphism. Wheneer α() = β() + r for some r R and all V (i.e., [α] = [β] ), we hae (g α)() = α( g) = β( g) + r = (g β)() + r, for all V. We later proe (Lemma 1.5.9) the action respects simplicial structure. Remark Recalling Obseration it is clear that for each g GL d (Q p ) mapping α g α stabilizes special norms, hence the action on building stabilizes ertices. 1.5 Matrix representaties In this thesis we shall inestigate the combinatorial properties of the building, hence we are interested in the combinatorial simplicial structure rather than the topological one. From now on we won t concern ourself with non-special norms. Using Obseration we will usually define ertices with Z p -lattices. In computations it will be worthwhile working with matrix data for ertices and apartments, which we are going to define. Using matrix representaties we will always assume we hae some basis F = {u 1,..., u d }, of V. The geometric meaning of all coordinates data is always related to basis F. Denote by E the canonical basis of V, that is, E = {e 1,..., e d }, where e i has 1 on i-th position and 0 s otherwise. Definition Let A = (A i,j ) d i,j=1 GL d (Q p ) has entries. Haing some basis F = {u 1,..., u d } of V = Q d p, denote by A :F ertex defined by Z p -lattice spanned by i = A i,1 u 1 + A i,2 u A i,d u d, i = 1,..., d. (1.7) And by A a:f we denote an apartment whose splitting basis is that from (1.7). That is, the ertices of A a:f are exactly {(DA), D Diag d (Q p )}. Following lemma concerns the ambiguity of just defined matrix representaties. Lemma If A, B GL d (Q p ), then A = B if and only if there is X GL d (Z p ) (i.e., det(x) Z p ) and integer i such that B = p i XA. Consequently, if P is permutation matrix and D Diag d (Q p ), then A a = (DP A) a. Proof. Clearly A = B if and only if Z p -lattices L = Row Zp (A) and K = Row Zp (B) define the same reduction class. That occurs if there is i Z such that L = p i K, which is equialent to existence of X GL d (Z p ) such that A = p i XB. 12

17 Let matrices P, D be as in the statement of the lemma. Gien any ertex α of apartment (DP A) a it has form (D 0 DP A), for some D 0 Diag d (Q p ). Denote by D 1 the diagonal matrix satisfying equality D 0 DP = P D 1 (the existence of such matrix is rather obious). Using the fact P GL d (Z p ), first part gies us (D 0 DP A) = (P D 1 A) = (D 1 A). So α belongs to A a. And by similar arguments eery ertex of A a belongs to (DP A) a. Definition Define relation of equialences and a on set of matrices in GL d (Q p ) by A B there are X GL d (Z p ), i Z, such that A = p i XB. A a B there are D Diag d (Q p ), P permutation matrix such that A = p i XB. Later we proe in Proposition that in the part concerning representaties of apartments conerse holds (i.e., A a B if and only if A a = B a ). In other words, gien some splitting basis u 1,..., u d of an apartment A, we can get all the other splitting bases by permuting the ector u i s and multiplying them by scalars. Obseration If ertex α belongs to apartment A, then there is matrix A GL d (Q p ) such that A = α and A a = A. Proof. Gien any representatie B of A, there is D Diag d (Q p ) such that α = DB. By Lemma also A = B a. Working with coordinates, a common trick is choosing a suitable basis F. For example, inestigating apartment A we choose basis F to be the splitting basis for F, hence A = I a:f. This is the reason we established matrix representaties with respect to arbitrary basis, not only the canonical one E. We hae to describe how the skipping between basis affect the matrix representaties. Lemma Let F 1 and F 2 be two bases of V and T be the matrix of transition between them (i.e., rows of T are ectors of F 2 with respect to F 1 ). For any A GL d (Q p ) we hae A :F1 = (AT 1 ) :F2 and A a:f1 = (AT 1 ) a:f2. Proof. Proing this is just easy exercise from linear algebra. In most of cases it will be clear with respect to which basis we use the matrix representaties. Hence, we will usually leae the reference to basis from the notion (e.g., ertices will be defined as A ). Remark Notice that matrix representaties of ertices are unique up to multiplication by certain matrices from left and skipping between two bases F 1 and F 2 is affects the matrix representaties by multiplication by transition matrix from right. Hence, from associatiity of matrix multiplication follows: A :F1 = B :F1 A :F2 = B :F2 and A a:f1 = B a:f1 A a:f2 = A a:f2. The essential knowledge we gain from following lemma, is that, neer mind which basis we choose, the GL d (Q p ) acts on matrix representaties by matrix multiplication from right. 13

18 Lemma Let ρ be the action from Definition Gien g GL d (Q p ), with respect to canonic basis E the action of g on ertices of B is this: ρ g : A :E (Ag 1 ) :E. If basis F arises from canonical basis E by transition matrix T (i.e., F is gien by rows of T ) then g acts on ertices of B as follows: ρ g : A :F (AT 1 g 1 T ) :F. Proof. Let α be special norm corresponding to Z p -lattice Row Zp (A) and g GL d (Q p ), then g α by definition acts on V as follows We hae α( g) : max{k Z g p k Row Zp (A)}. g Row Zp (A) Row Zp (A g 1 ). Hence special norm g α corresponds to Z p -lattice Row Zp (A g 1 ), this gies us the first part. If α = A, then with respect to standard basis α = (AT 1 ) :E and so ρ X (α) = (AT 1 X 1 ) :E. Skipping back to basis F we get ρ X (α) = (AT 1 X 1 T ) :F. Lemma Vertices α 0,..., α k form an k-simplex if and only if there are matrices A 0,..., A k such that (A i ) s are all the ertices α i s and Row Zp (A 0 ) Row Zp (A 1 ) Row Zp (A k ) prow Zp (A 0 ). (1.8) Proof. There has to be chain of lattices like in (1.6) witnessing the k-simplex. We take for matrices A i s the matrices whose rows generates the lattices in sense of Definition Lemma GL d (Q p ) acts on B by simplicial automorphisms (with respect to abstract simplicial structure). Proof. Let ertices α 0,..., α k form k-simplex. Then by Lemma there are matrices A 0,..., A k GL d (Q p ) satisfying (1.8). It is rather easy to show that chain (1.8) is equialent to Row Zp (A 0 X) Row Zp (A 1 X) Row Zp (A k X) prow Zp (A 0 X), for any X GL d (Q p ). Using Lemmas and we see that ertices ρ g (α 0 ),..., ρ g (α k ) form k-simplex. Consequently, action ρ respects the whole abstract simplicial structure. 14

19 2. The case of SL 2 (Q p ) In this chapter we describe the simplicial structure of B(GL(2, Q p )), to which we will refer in this chapter as B. According to Lemma it is 1-dimensional, i.e., a graph. The ertices correspond to reduction classes of special norms. By the reduction apartments are 1-dimensional, and so they correspond to paths infinite on both sides (we will later show that bijectiely in Lemma 2.2.4). Let F be any basis of V, with respect to which we will use matrix representaties (recall Definition 1.5.1). We will be often changing F to make some just inestigating ertex hae nice matrix representatie. Define ertex ) α = ( (the letter α will hae this meaning in the whole chapter). Let β be any other ertex gien by some Z p -lattice L. By Lemma there is matrix representatie ( ) c d β =, e f where c, d, e and f are elements of Z p and minimum among ν(c), ν(d), ν(e) and ν(f) equals zero, say it is ν(c) (i.e. c Z p ). According to Lemma elementary row operations on representaties do not change ertex. Hence ( ) ) ) c d β = e f = ( c d 0 f ec 1 d = ( c d 0 p ν(f ec 1 d) In first step we add ( ec 1 )-times first row to the second row. Define k = ν(f ec 1 d). Following sequence is walk from α to β: ( ) ( ) ( ) c d c d c d α = p 0 p k = β. (2.1) The determinant of the first matrix equals c Z p, hence the matrix belongs to GL(Z p ) and represents α by Lemma Using Lemma all consecutie ertices are connected by an edge. In computation aboe we assumed ν(c) = 0, howeer all other cases can be treated similarly. We proed that each ertex of B has some apartment in common with α. Since α can be any ertex (by choose of basis F) we obsere: Obseration In B(GL 2 (Q p )) eery two ertices hae some apartment in common, consequently the building is connected graph. Remark Just finished computation demonstrates typical treating with matrix representaties: Gien two ertices α and β choose conenient basis F such that α = I and then find matrix B, matrix representatie of β, suitable to our aim. We know that each ertex has it splitting basis (eery lattice L V has Z p - basis). Transiting to building notion: eery ertex belongs to some apartment. By Definition condition iii the splitting basis should exist for eery pair of simplices. We conclude the introduction to this chapter with weak ersion: 15.

20 Lemma Eery edge {β, γ} leis on some apartment. In detail: there is basis 1, 2 such that β and γ are gien by lattices Z p 1, 2 ) and Z p 1, p 2. Equialently: β and γ hae representaties with c, d, e, f Q p. ( ) ( ) c d c d β =, γ =, (2.2) e f pe pf Proof. Recall that changing basis F changes matrix representatie by multiplication by transition matrix from right. This clearly preseres the relation between matrices in (2.2). Hence it is enough to proe the lemma for any choose of F,. So choose F to be splitting basis of apartment common to β and γ (which exist by Obseration ) and β = α = E. Hence γ has by definition diagonal matrix representatie ( ) c 0 γ =. 0 d By Lemmas and there is k Z such that γ = diag(p k c, p k d) and Row Zp (I) Row Zp (diag(p k c, p k d)) prow Zp (I). (2.3) It follows 0 ν(p k c) 1, 0 ν(p k d) 1 and among ν(p k c), ν(p k d) are both 0 and 1 (otherwise the strict inclusions would not holds). Suppose ν(p k d) = 1 and ν(p k c) = 0, then 1 = u 1, 2 = u 2 is the basis we are looking for, since Z p 1, p 2 and Z p p k c 1, p k d 2 ) are the same lattice. In the case ν(p k d) = 0 and ν(p k c) = 1 by similar argument 1 = u 2, 2 = u 1 is the basis we are looking for. 2.1 Parametrization of apartments passing through the origin ertex α Now let us describe all apartments passing through the origin ertex α = ( Denote by A any such apartment. Condition α A ensures existence of matrix representatie A gien by ( ) c d A =. e f such that α = A (Obseration 1.5.4). Clearly we can choose A so that ) Row Zp (I) = Row Zp (A) (by multiplication it by suitable power of p). Such A hae elements in Z p and det(a) = cf ef Z p. It follows A has in each row and each column element from Z p. Hence we can multiply its rows and eentually interchange them (that. 16

21 is by Lemma 1.5.2allowed for representaties of both ertices and apartment) to get representatie ( ) 1 d A =, (2.4) c 1 where c, d Z p and 1 cd Z p. Definition For c, d Z p with ν(cd 1) = 0 denote by A c,d the apartment gien by (2.4). We hae just parametrized all apartments passing through α. The parametrization is not unique: clearly for c, d Z p we hae ( ) ( ) ( ) 1 d d c 1 A c,d = = c 1 1 c 1 = d 1 = A 1 d 1,c 1. (2.5) a a Our aim in the following text is proing that this is the only ambiguity. Passing through the apartment haing 1, 2 as splitting basis is possible in two directions: one corresponds to multiplying the 1 by powers of p and the other corresponds to the same for 2. The reduction causes that these two operations could be read as going forward and back: Z p ( 1, 2 ) Z p (p k 1, 2 ) Z p (p k 1, p k 2 ) = p k Z p ( 1, 2 ). The first and the second lattice define the same ertex. It is clear that ertices Z p ( 1, p i 2 ) and Z p ( 1, p i+1 2 ) are connected with an edge. In this sense apartments corresponds to walks (in fact to paths - Lemma 2.2.4). Definition For an apartment A c,d call the sub-graph A c,d generated by ertices ( p i p i ) d, c 1 i N 0 its left wing and call the ertex corresponding to i the i-th ertex of A c,d. And the sub-graph A c,d generated by ertices ( 1 d p i c i N 0 we will call its right wing and the ertex corresponding to i the i-th ertex of A c,d. Further, for the whole building B, define the left wing of the building B to be the sub-graph of B generated by all the left wings and analogously define B. In Lemma we will proe that each apartment is a path (or a line if we consider only the edges), and the left and right wings are half-lines. According to Obseration eery ertex has some apartment in common with α, hence eery ertex belongs to a left or right wing. Similarly as eery ertex is contained in seeral apartments some ertices are placed in both left and right wing. Of course the concept of left and right wings is not well defined for the building, since it depend on fixed basis - the place we are watching the building. p i 17 ), a

22 Lemma Let c, d Z p, such that cd 1 Z p. Then A c,d is identical with A c,0 and A c,d is identical with A 0,d. Proof. At first note that A c,0 is well defined, since c 0 1 = 1 Z p. We will proe that the i-th ertex of A c,d is the i-th ertex of A c,0. That follows from this computation: ( p i p i d c 1 ) ( p i (1 cd) 0 c 1 ) ( p i 0 c 1 At first step we deduced the first row by p i d times the second row. At the second step we multiplied the first row by (1 cd) 1. The inerse is element of Z p by Definition Both these operations are alid for matrix representations of ertices. The case of right wing is treated using analogous computation. Hence the left wing A c,d only depends on c, and the right one A c,d on d. Concerning left wings it is enough to work with A c,0. Lemma Let c = i=0 c i p i, where each c i belongs to {0,..., p}. Then the k-th ertex of A c,0 (counting α as the 0-th) depends only on c 0,... c k 1. Two wings A c,0 and A c,0 share their 0-th to ν(c c )-th ertex and then they split. Similarly in the case of right wing A 0,d and any pair of right wings A 0,d and A 0,d : first k ertices of A 0,d only depends on d (mod p k ), and A 0,d and A 0,d share its 0-th to nu(d d )-th ertex Proof. We will proe the case of left wings, the other is analogous. Let us inestigate under which circumstances two wings A c,0 and A c,0 share some ertex. As (i, j) pass through N 0 N 0, ( ( ) ( ) ) p i 0 p j 0 c, 1 c (2.6) 1 pass though all ordered pairs (ertex of A c,0, ertex of A c,0). The two matrices (A i, B j ) defined by (2.6) represent the same ertex if and only if there is X GL(2, Z p ) (that is det(x) Z p ) and k Z, such that p k XA i = B j, as follows from Lemma Concerning aluation of determinants there has to be 2k i = 2k + ν(det(x)) + ν(det(a i )) = ν(det(b j )) = j, hence 2k = j i. Direct computation (using 2k = j i) leads to ( p k p k ) ( ) 0 p i 0 p k i (c c ) p k c = 1 ( p = j i ) ( ) ( ) 0 p i 0 p j 0 p i (c c ) 1 c = 1 c. 1 This equation determines X uniquely and the condition for X haing elements in Z p implies that k and k both hae to be 0, and so k = 0. Consequently i = j and c c (mod p i ). It follows that the k-th ertex of A c,0 only depend on c (mod p k ), that gies the first part. 18 ).

23 Remark From Lemma follows that the ertices in a left wing are in one-to-one correspondence with finite sequences of elements from {0, 1,..., p 1} (and similarly for the right wing). And the sequence corresponding to ertex codes the path leading from α to it. Lemma Two wings A c,0 and A 0,d hae in common exactly their 0-th to ν(cd 1)-th ertex. Proof. Similarly as in the preious Lemma let s inestigate when there is a ertex common to A c,0 and A 0,d. For ( ) ( ) p i 0 1 d A i =, B c 1 j = 0 p j, (2.7) i, j N 0, we are looking for X GL 2 (Z (p) ) and k Z such that p k XA i = B j. Again we must hae 2k = j i. Matrix X is uniquely gien by following computation ( p k p k c p k+i ) ( ) p i 0 = p (k+i) (1 cd) dp k ( p = j i c p j ) p i (1 cd) d ( p i 0 c 1 ) = c 1 ( 1 d 0 p j Consider two possible cases: at first let ν(c) > 0 or ν(d) > 0, then ν(1 cd) = 0 and there must be both k + i 0 and (k + i) 0, consequently 2i = 2k = j i j = i, hence i = j = 0 since i, j are by definition non-negatie. This case leads to triial solution. Next suppose c, d Z p, than clearly k, k 0 and consequently i = j and cd 1 (mod p i ). Description of all apartments passing through the origin α is a bit harder since two apartments A c,d and A c,d are also identical if A c,d = A c,d and A c,d = A c,d. Notice that Lemma for c Z p haing ν(c) 1 we hae ν(cd 1) = 0 for any d Z p. Hence by Lemma no ertex except the 0-th α belongs to B. Considering Lemma we obsere: Obseration Vertex β lying in some left wing A c,d (right wing A c,d, resp.) which is not equal to α is element of B B if and only if c Z p (d Z p, resp.). Following lemma ensures that two apartments passing through α can hae non-triial intersection of either types: left wing with right wing and right wing with left wing or left wing with left wing and right wing with right wing. Lemma Let A c,d be apartment and A c,0 some left wing. Then at least one of following graphs is triial (equals {α}): A c,d A c,0 = {α} or A c,d A c,0 = {α}. And similarly for any right wing A 0,d. Furthermore let A e,f is triial (equals {α}) ( Ac,d ) ( A e,f A c,d ) A e,f be any other apartment. Then one of following graphs or ( A c,d ) ( A e,f Ac,d A e,f ). ). 19

24 Proof. Suppose that A c,0 has more then two ertices in common with both A c,d and A c,d, then by Lemmas and c c (mod p) and c d 1 0 (mod p). Hence 0 cd 1 c d 1 0 (mod p), which in contradiction with Definition If A c,d A e,f {α} then applying the already proed part for the apartment A c,d and A e,f = A e,0, we get A c,d A e,f = {α}. An using first part for apartment A e,f and A c,d = A 0,d we get A c,d A e,f = {α}. So the second graph is triial. Wheneer any of the four intersections is non-triial the two intersections corresponding to the other graph are triial. Lemma Apartments A c,d and A e,f are identical if and only if either c = e and d = f, or c = d 1 and d = e 1. Proof. By Lemma we can hae A c,d = A e,f if and only if either A c,d = A e,f and A c,d = A e,f or A c,d = A e,f and A c,d = A e,f. Using Lemmas and the first case implies c = e and d = f, the second one implies c = d 1 and d = e 1. Lemma Two apartments A = A and A = A are identical if and only if there is permutation matrix P and diagonal matrix D, both elements of GL 2 (Q p ) such that A = DP A. Proof. The if part we already proed in Lemma It is enough to proe the only part for any choice of basis F (see Remark 1.5.6). If the apartments A and A are identical, they surely share some ertex, choose F such that the shared ertex is the origin α. The matrices A and A can be transformed to Ā and Ā of form (2.4) with c, d and c, d, by the same way as we described in the beginning of this section. Notice, we used there only multiplication from left by the allowed matrices P s and D s. According to preious Lemma either holds: c = c and d = d (in this case we are done) or c = d 1 and d = c 1. In the later case we get Ā from Ā by the same computation as in (2.5). 2.2 Description of the graph structure of the building B(GL 2 (Q p )) Lemma Eery apartment is an infinite path without ends. Proof. It is enough to proe the statement for apartments passing through α. For A c,d we would like to show that the mapping Z i ( 1 d is injectie. We already know that it maps consecutie integers to neighbours. Using Lemma we see the mapping is injectie on both Z 0 and Z 0. And Lemma implies the images of Z >0 and Z <0 are disjoint, since ν(cd 1) = cp j p j )

25 Lemma Eery ertex of the building B(GL(2, Q p )) is of degree p + 1. Specially in the case of α, p of its neighbours belong to B and p of its neighbours belong to B. Proof. Let β, α be edge. According to Lemma there is an apartment A containing the edge β, α. Since the apartment is union of its left and right wing β has to be the first ertex of some wing. If the wing is left of form A c,0 (the Lemma ensure we can assume this form), the ertex β depends according the Lemma only on c (mod p), that gies p possibilities and so does the right wing case. Howeer for c Z p there is A c,0 = A 0,c 1 as states the Lemma 2.1.6, hence p 1 ertices we got form B are identical with some ertices we got from B. That gies 2p (p 1) = p + 1 neighbours. Since we chose α arbitrarily, all ertices are of degree p + 1. Proposition Let ω be infinite walk in building B, that is a sub-graph of form, ω 2, e 2, ω 1, e 2, ω 0, e 2, ω 1, e 2, ω 2,, where the ω i s are ertices and e i s edges, such that e i connect ω i and ω i+1, for each integer i. If for each i Z we hae ω i 1 ω i+1, then there is apartment, whose simplicial structure is exactly the walk ω. Proof. Choose basis such that α = ω 0. Vertex α has according to Lemma p + 1 neighbours and p of them lay in B, denote them γ 1,..., γ p. Clearly at least one of ω 1, ω 1 leis in B, WLOG suppose it is ω 1. Claim: For any integer n there is c n, element of Z p, such that ω 0, e 1, ω 1, e 2, e n, ω n, (2.8) is the initial segment of A c n,0. Proof of the Claim: Since eery ertex is by Lemma of degree p+1, there are exactly p n possibilities for walk (2.8) with ω 0 = α and ω 1 {ω 1,, ω p } (proing this is easy exercise). On the other hand, according to Lemma 2.1.4, there are exactly p n paths of length n haing beginning at α, which are initial segment of some left wing A c,0. Eery such path surely contains some of γ i s as its first ertex (recall α is counted as the 0-th) and since it is path it respects the condition ω i 1 ω i+1. Comparing the cardinalities we get the Claim. Using the Claim we obtain sequence {c n } n N. Let n, m N, n < m, then A c n,0 and A c m,0 shares its 0-th to n-th ertices, hence c n c m (mod p n ), that follows from Lemma Consequently {c n } n N is Cauchy sequence, with respect to p-adic topology and there exist c = lim n c n. Now define the wing A c,0, which follows ω 0, e 1, ω 1, e 2 ω 2,, since it share with it arbitrarily long initial segment. If it is possible to construct the right wing A 0,d following the other direction of ω we are done: ω 1 ω 1 hence ν(cd 1) = 0 as Lemma states. That implies A c,d is well defined and from Lemma it is the apartment we are looking for. 21

26 B B B \ B B \ B ( ) ( ) ( ) ( ) ( ) ( ) ( ) Figure 2.1: Essential part of building for group SL 2 (Q 2 ). If the construction of right wing is impossible (ω 1 does not belongs to B), we can construct again left wing A d,0 as before. Since c d (mod p) (otherwise they would share their first ertices) one of c, d has to be inertible, say it is c. We hae well defined A 0,c 1, which is identical to A c,0 as Lemma asserts. Similarly as in the preious case we are done with the apartment A d,c 1. Corollary Infinite paths without ends correspond to apartments of building B. Proof. Since eery infinite path without ends respects the condition in statement of Proposition 2.2.3, we can find the corresponding apartment. Lemma ensures the other direction. Corollary The graph structure of B has no cycles. Proof. Cycles are walks that respect condition in statement of Propposition Hence, if there was cycle, we would be able find an apartment, that would infinitely run around the cycle. Howeer that is impossible, since apartments are infinite paths, as Lemma states. At this point we can make a full picture of the building B. Because the graph is acyclic, each wing is generated by the addressing paths as Remark describes. The whole graph is composed from two complete p-ary trees haing α as root and being glued together ia identifying p 1 of their branches together ( A c,0 = A 0,c 1). See Picture 3.1. Obseration Denote the neighbour ertices of α by: ( ) ( ) ( ) p 0 p 0 1 c 1 γ 0 =, γ 0 1 c = =, γ c 1 0 p p = ( p where c = 1,..., p 1. The graph B after deleting α splits into p + 1 disjunct branches: one in B B (with root ertex γ 0 ), p 1 of them are contained in B B (with root ertices γ1,..., γ p 1 ), and the last one in B B(with root ertex γ p ). 22 ),

27 We can now proe one of axioms of buildings: Definition condition (iii). Theorem In B eery pair of simplices hae some apartment in common. Proof. Enough to proe the worst case, when the simplices are edges. Since the graph is connected by Obseration , it is easy combinatoric exercise to show, that any pair of edges can be completed into path. This path could be on both sides extended arbitrarily long, because in building B is no ertex of degree 1 and no cycle, that assert Lemmas and Hence the edges lay on some infinite path without ends and we can conclude using Lemma Some formulas Since the graph B is acyclic and apartments are paths, to compute the graph distance of two ertices, we can take any apartment( containing ) them and compute c d the distance on this apartment. For any matrix, haing elements in e f Q p define ( ( ) ) c d c d := ν det, e f p e f the aluation of its determinant. Theorem For any pair of ertices ( ) ( c d r s β =, γ = e f t u ) their graph distance equals { c d r s + 2 min e f p t u p c r d s, p c t d u, p e r f s, p e t f u }. p Proof. At first note, that the formula is inariant under change of basis: Since change of basis affects the formula by multiplying all the matrices by a transition matrix T from the right (Lemma 1.5.5), each of the first two summands increases by T p. Howeer, that is balanced by decreasing of the third summand by 2 T p. Next see that scalar multiplication of any of the two matrix representaties by x Q p does not affect the alue of the formula either. Hence we can choose basis so that ) γ = ( and salary multiply the matrix representaties of β to make ( ) c d β =, e f with c, d, e, f Z p and one of them equal to 1. Changing the rows of matrix representaties clearly preseres alue of the formula, hence we can suppose the 1 is c or d. These both cases can be treated similarly, we will proceed with c = 1. 23

28 The only non-anishing summand is 1 d e f, because p = 0 and the p last summand shows to be min{ν(c), ν(d), ν(e), ν(f)}. Since c, d, e and f are elements of Z p and ν(c) = ν(1) = 0, the minimum is zero. From ( ) 1 d = e f we see, that ertex β lies in the apartment ( ) ( ) 1 d 1 d 1 0 from = = γ. e f p ( 1 d 0 f de ( 1 d 0 1 ) ) a in distance ν(f de) = Lemma Let A is an apartment, then it has matrix representation of form ( 1 g 1 h ) or a ( 1 g h 1 ) or a ( g 1 h 1 ), (2.9) a with g, h Z p. If A is one of matrices in (2.9), then the ertex A is the closest ertex to the α = E among all the ertices of the apartment A. Proof. Gien any matrix representatie A of A, multiply the rows of A by conenient scalars to make the entries elements of Z p and one equal to 1. Then eentually interchange the rows. These operations are allowed for matrix representaties by lemma Let us proe the second part for case A = ( 1 g 1 h ). a Using Theorem compute the distance of ertices α = ( ) and β = ( p i p i g 1 h ) A, where i Z: p i p i g min{ν(p i ), ν(p i g), ν(1), ν(h)} = 1 h p = i + 1 g 1 h 2 min{0, i} = p 1 g 1 h + i. p Lemma Let A c,d, A e,f be two apartments. Then the number of edges common to: (i) A c,d and A e,f equals ν(c e), (ii) A c,d and A e,f equals ν(d f), 24