aa r X i v : . [ m a t h . G R ] S e p Affine Λ Buildings II:A reduction of axioms
Curtis D. BennettLoyola Marymount UniversityOctober 14, 2018
Jacques Tits introduced the notion of a building as a geometry associatedto groups of Lie type in [T1], providing new geometries associated to theexceptional groups of Lie type. In 1972, F. Bruhat and Tits [BT] developed atheory of affine buildings for the purpose of studying groups over fields havinga discrete valuation, although their work applied more generally to groupsover fields having a valuation over the real numbers. Affine Λ-buildings werefirst introduced by the author in [B1] and [B2] as generalizations to theBruhat-Tits buildings, allowing for groups over fields having a valuation intoany totally ordered abelian group Λ (and also generalizing the notion of aΛ-tree). Recently, Linus Kramer and Katrin Tent have made use of affineΛ-buildings in their study of asymptotic cones and their short proof of theMargulis conjecture [KT], [KST].In [B2], the author defines an affine Λ building as a pair (∆ , F ) satisfy-ing a set of six axioms given in section 2. The first four of theses axiomsare relatively easy to check in most cases. The sixth axiom is a little lessstraightforward and the sixth axiom can be particularly difficult to show.Consequently, to make affine Λ-buildings more useful as a tool, it is worth-while to find an easier set of axioms with which to work. The purpose of thispaper is to provide an easier set of axioms by extending results of Anne Par-reau [P] on the equivalence of axioms for Euclidean buildings. In particular,in section 3 we define a strong exchange condition, mimicking Proposition1.27 from [T2]. In Section 4 we then prove that a pair (∆ , F ) is an affine Λ-building if and only if it satisfies the first four axioms together with the strongexchange condition. More recently, Petra Schwer has used these results toobtain an even stronger conclusion [S]. In the first part of this section we follow [B2] to give the definition of an affineΛ-building. In the second part of this section we give precise definitions ofour new replacement axioms. Λ -apartment Σ Let Λ denote a totally ordered Abelian group viewed as a Q -module. Let W be a spherical Coxeter group, Φ the associated spherical root system withbase { α , . . . , α n } , and let A = ( h α i , α j i Φ ) be the associated Cartan matrix,and let D = Diag( d , . . . , d n ), be given so that D − A is symmetric. DefineΣ = ( n X i =1 λ i α i | λ i ∈ Λ ) ∼ = Λ n . We represent elements of Σ by the coordinates ( λ , . . . , λ n ), and note thatany sum of roots with coefficients in Λ corresponds to a unique element ofΣ. Moreover, using this, we extend the action of W to Σ linearly. Letting T be the set of Λ-translations of Σ normalized by W , we define the affine Λ -Weyl group to be the group W = T W . Reflections of W are defined to beconjugations of reflections of W .Walls of Σ are defined to be the fixed points of reflections of W . For v ∈ Σand a wall M associated to the reflection over α i , using Proposition 2.1 of[B2], we have v can be written uniquely as v = m + λ i α i for some m ∈ M and λ ∈ Λ. For each i = 1 , . . . , n , we let v i = d i λ i , and in general, we let v w ( α i ) = ( w − ( v )) i for w ∈ W (this is well defined by Corollary 2.5 of [B2]).As is shown in [B2], x ∈ Σ is uniquely determined by the n -tuple ( x i , . . . , x n ).A half-apartment of Σ is any of the sets H w = w ( { v ∈ Σ | v i ≥ } ) , fundamental sector of Σ is the set b S = { v ∈ Σ | v i ≥ i = 1 , . . . , n } . A set S is a sector of Σ if S = w ( b S ) for some w ∈ W , and we say that S is based at w ( o ) where o = (0 , . . . , ∈ Σ. The fundamental sector panel oftype i of b S is the set P i = { v ∈ b S | v i = 0 } , and we again use W to define the sector panels of type i of any sector.We now define a Λ-distance function on Σ as follows. Let α i be a funda-mental root of Φ and v = ( λ , . . . , λ n ). Define( α i , v ) = d i a ij λ j where A = ( a ij ) is the Cartan matrix associated to Φ, and D = Diag( d , . . . , d n )is the given “symmetrization” factor of A (as above). For α ∈ Φ, we define( α, v ) by extending the function linearly. The Λ-distance between vectors v , v ∈ Σ is given by d ( v , v ) = X α ∈ Φ + | ( α, v − v ) | , where | | denotes the usual absolute value function on Λ. As shown in [B2], d is a symmetric W -invariant function that satisfies the triangle inequality,and so it is appropriate to call d a Λ-metric.With these definitions, we define two sectors to be parallel if they areat bounded distance from each other. (Note that it follows that sectors areparallel if and only if their intersection contains a subsector.) We similarlydefine parallelism of sector-panels, and in [B2] it is shown that a sector orsector-panel X is parallel to X if and only if X and X are translations ofeach other.A set Ω ⊂ Σ is convex if it is the intersection of half-apartments, and itis closed and convex if it is the intersection of finitely many half-apartments.
Definition:
Given a sector S of Σ based at y , we say that a subset V ofΣ contains the sector-germ S y of S if there exists an open set U of ∆ suchthat U ∩ S ⊆ V and y ∈ U ∩ S . Similarly if P is a panel of S , we say that V contains the sector-panel germ P y if y ∈ U ∩ P y ⊂ V .We say that the sectors S = w ( S ) and S = w ( S ) are i -adjacent if w = r w , in W = W/ T , and we define a sector-gallery to be a sequence S , . . . , S k
3f sectors such that S i − is j i -adjacent to S i . The type of the sector-gallery is j j . . . j k . The sector distance is then given by d ( S , S k ) = r j k r j k − . . . r j r j .Given two sector germs S y and T y of Σ (corresponding to sectors S and T ),the sector germ distance from S y to T y is given by δ ( S y , T y ) = d ( S, T ).We note here that if Σ were the simplicial affine apartment, a sector germcorresponds to the chamber at the base of the sector. The distance betweentwo sector germs based at the same point corresponds precisely with thedistance between those two chambers in the residue of the point. On theother hand, the distance between two sectors corresponds to the distancebetween the parallel classes of those sectors in the apartment at infinity.We conclude the discussion of the Λ-apartment with a straightforwardlemma that will be used in the sequel
Lemma 1
Let S be a sector of Σ based at x . If S ′ is a subsector of S , thenthe convex hull of S ′ and x is S . We refer the reader to [B2] for any other definitions needed. Λ -building In this section we will state the various axioms that we will use in this paper.The first set of six axioms are those in the original definition of an affineΛ-building.Retaining the notation of the previous section, let ∆ be a set, Φ a sphericalroot system, Σ the associated canonical Λ-apartment, and F a set of mapsfrom Σ to ∆. An apartment of ∆ is a set f (Σ) for some f ∈ F . We definethe sectors, sector-panels, walls, half-apartments, etc. of ∆ to be the imagesof the same such under any f ∈ F .The pair (∆ , F ) is an affine Λ -building if the following conditions aresatisfied:(A1) Given f ∈ F and w ∈ W , then f ◦ w ∈ F .(A2) (Compatability Axiom) Given f , f ∈ F , if f (Σ) ∩ f (Σ) = ∅ then f − f (Σ) is a closed convex set of Σ, and there exists w ∈ W such that f | f − f (Σ) = f ◦ w | f − f (Σ).(A3) Given x, y ∈ ∆ then there exists f ∈ F such that x, y ∈ f (Σ). (Notethat axioms (A2) and (A3) imply that the metric d on Σ extends via F to a well defined distance function d : ∆ × ∆ → Λ.)4A4) (Subsector axiom) Given sectors S , S ⊂ ∆, then there exists subsec-tors S ′ ⊂ S and S ′ ⊂ S and f ∈ F such that S ′ ∪ S ′ ⊂ f (Σ).(A5) (Retraction axiom) Given f ∈ F and x ∈ f (Σ), there exists a retraction ρ : ∆ → f (Σ) such that ρ − ( x ) = x . (Note that this axiom implies thedistance function d on ∆ satisfies the triangle inequality (see [B2]).)(A6) ( Y -condition) Given three maps f , f , f ∈ F such that f i (Σ) ∩ f j (Σ)is a half-apartment for i = j , then f (Σ) ∩ f (Σ) ∩ f (Σ) is non-empty.Given this definition of a Affine Λ-building (∆ , F ), the set of apartmentsof (∆ , F ) is given by A = { f (Σ) | f ∈ F } . Abusing notation, we will often refer to the set ∆ as satisfying the axiomsand assume the set A of apartments (and maps) is given. In this section, we present two new axioms and prove their equivalence to(A6) (given (A1)-(A5)). Our two new axioms are(EC) (Exchange Condition) Given two maps f , f ∈ F such that f (Σ) ∩ f (Σ) is a half apartment, then there exists a map f ∈ F such that f (Σ) ∩ f j (Σ) is a half apartment for j = 1 ,
2. Moreover, f (Σ) is thesymmetric difference of f (Σ) and f (Σ) together with the boundarywall of f (Σ) ∩ f (Σ).Note that the exchange condition can be restated in “apartment language” as:Given two apartments A and A of (∆ , F ) intersecting in a half-apartment H with boundary wall M , then ( A ⊕ A ) ∪ M is also an apartment (where ⊕ denotes the symmetric difference).(SE) (Strong Exchange Condition) Suppose f ∈ F and S is a sector of(∆ , F ) such that P = S ∩ f (Σ) is a sector-panel. Letting M be thewall of f (Σ) containing P . Then there exist f , f ∈ F such that f (Σ) ∩ f j (Σ) is a half-apartment and ( M ∪ S ) ⊂ f j (Σ) (for j = 2 , A of ∆ and a sector S with a sector-panel of S lying in A , then there exists5partments A and A such that S ⊂ ( A ∩ A ) and A ∩ A j is a half-apartmentcontaining the sector-panel of S for j = 1 , S and S of ∆ to be parallel if S and S are at bounded distance fromeach other. We similarly define sector panels to be parallel (see [B2] for howwe define the type function on sector-panels) if they are at bounded distance.We note that axioms (A2) and (A4) imply the sector-distance function onΣ extends to a well-defined W -distance function d on the parallel classes ofsectors of ∆.Let ∆ (= (∆ , F )) be an affine Λ-building of type W and define ∆ ∞ = { S ∞ | S is a sector of ∆ } . We say that S ∞ and S ∞ are i -adjacent if S and S have parallel sector-panels of type i . For an apartment A of ∆, we definethe set A ∞ = { S ∞ | S is a sector of A } . Then if ∆ is an affine Λ-building oftype W , the set A ∞ is a thin chamber complex of type W . Theorem 2 ([B2, Theorem 3.7]) Let (∆ , F ) be a pair satisfying conditions(A1)-(A5). Then the chamber system ∆ ∞ is a spherical building of type W with apartments in one-to-one correspondence with the apartments of ∆ .Similarly, the walls and panels are in one-to-one correspondence with theparallel classes of walls and the parallel classes of sector panels of ∆ . In [B2], the assumptions are actually that (∆ , F ) satisfies condition (A6)also, but an analysis of the proof shows that condition (A6) is never used.In fact, the Y -condition (A6) is used in [B2] primarily to avoid pathologicalcases and to force the existence of Λ-trees associated to the walls and panelsat infinity as in [T3].Given Thoerem 2, we are now ready to prove the first of our equivalenceresults. Proposition 3
Given (∆ , F ) satisfies conditions (A1)-(A5), then condi-tion (A6) is equivalent to condition (EC). Proof:
Let us begin by assuming that (∆ , F ) satisfies (A6), and suppose A = f (Σ) and A = f (Σ) are two apartments of ∆ with A ∩ A = H ahalf-apartment. Then A ∞ and A ∞ are apartments of ∆ ∞ that intersect in ahalf-apartment. By spherical building theory, it follows that there exists anapartment A ∞ whose chambers are the chambers of A ∞ ⊕ A ∞ . Theorem 2now implies that there exists an apartment A of ∆ corresponding to A ∞ .6onsequently by (A6) A ∩ A ∩ A is non-empty. Since A ∞ ∩ A ∞ is a half-apartment and A ∩ A is convex by condition (A2), it follows that A ∩ A isa half-apartment. Similarly A ∩ A is a half-apartment. Condition (A6) nowimplies that A ∩ A ∩ A contains some element x ∈ ∆. Since x ∈ H and A ∞ contains the chambers of A ∞ not in A ∞ , it follows that A contains A − H .Similarly A − H ⊂ A . By convexity ∂H ⊆ A . But now the convexity of A implies that x ∈ ∂H (the boundary of H ) as otherwise the wall parallelto ∂H through x would not separate points of A ∩ A and A ∩ A . Thisimplies that (EC) holds.Now assume (A1)-(A5) and (EC) are all satisfied, and let A , A , and A be half apartments of ∆ such that any two intersect in a half-apartment.By way of contradiction, suppose A ∩ A ∩ A = ∅ . Let H ij = A i ∩ A j for i, j ∈ { , , } . Since H , ∩ H , = ∅ , it follows that if H is a half-apartmentof A with H , ∩ H ⊆ ∂H , then H , ∩ H is again a half-apartment. Now (EC)implies that there exists an apartment A such that A = ( A ⊕ A ) ∪ ∂H , .Note that H , ⊆ A , so that A ∞ consists of the same sectors as A ∞ . However,by Theorem 2, the apartments of ∆ are in one-to-one correspondence withthe apartments of ∆ ∞ . Therefore, A = A .We now prove the similar result for condition (SE). Proposition 4
Given (∆ , F ) satisfies conditions (A1)-(A5), then condi-tion (EC) is equivalent to condition (SE). Proof:
Suppose (∆ , F ) satisfies conditions (A1)-(A5) and (EC). Suppose A is an apartment and S is a sector such that S ∩ A is a sector panel of S . Again, by Theorem 2, in ∆ ∞ , A ∞ ∩ S ∞ is a panel. Therefore, for thebuilding at infinity, there is an apartment A ∞ of ∆ ∞ such that S ∞ ∈ A ∞ ,and A ∞ ∩ A ∞ is a half apartment. Let A be the corresponding apartmentof ∆. Since A ∞ ∩ A ∞ is a half-apartment of ∆ ∞ , it follows that A ∩ A is a half-apartment of ∆. We now apply condition (EC) to conclude theargument.Conversely, suppose (∆ , F ) satisfies conditions (A1)-(A5) and (SE), andlet A and A be apartments of ∆ intersecting in a half-apartment H . Let S be a sector of A such that S ∩ A is a sector-panel P of S , and let M be thewall of A containing P . By (EC), there exists an apartment A containing M such that A ∩ A is a half-apartment and A ∩ A is a half-apartment (as A must be one of the apartments guaranteed by (EC)) containing M ∪ S .By convexity, it follows that A = ( A ⊕ A ) ∪ M as desired.7e have now shown that the Y -condition can be replaced by either ofthe exchange axioms. In this section we prove our two main theorems. The first theorem pro-vides a general proof for affine buildings (Λ- or otherwise) that for any twosector-germs of ∆ are contained in a common apartment. That is, our resultgeneralizes the work of Anne Parreau [P] on Bruhat-Tits buildings, althoughour proof takes a different approach.The second theorem states ∆ is an affine Λ-building if ∆ satisfies axioms(A1), (A2), (A3), (A4), and (SE). The proof of the second theorem uses aslightly stronger result than the first theorem.To begin with we need a preparatory lemma.
Lemma 5
Let ∆ satisfy conditions (A1)-(A4) and (SE). If S and T aresectors of ∆ based at y , there exists an apartment A of ∆ such that S y ∪ T ⊂ A . Moreover, ℓ ( δ ( S y , T y )) ≤ ℓ ( d ( S ∞ , T ∞ )) and equality holds if and only if S and T are contained in a common apartment. By axiom (A4) there exists an apartment A ′ containing subsectors S ′ of S and T ′ of T , and d ( S ′ , T ′ ) = d ( S, T ). Consider the sector-gallery S ′ = S , . . . , S n = T ′ , and let A be an apartment containing S (and hence S ′ ). Let j be minimalsuch that S j +1 contains no subsector in A . If j = n exists (i.e., T ′ has asubsector T ′′ contained in A ), then T ′′ is a sector of A , and as y ∈ A ,by convexity it follows that T ⊂ A and there is nothing to prove. We willinduct on n − j . The basis step having been proven, assume S j +1 has nosubsector contained in A but S , . . . , S j all have subsectors in A . In thiscase, there exists a sector S ′ j +1 parallel to S j +1 (in A ′ ) such that S ′ j ∩ A is asector-panel (parallel to a sector panel of S j ). By condition (SE) there existsan apartment A j +1 containing S ′ j +1 and the sector germ S y (since for anywall S y must lie on one side or the other of the wall). If S is contained in A j +1 , then we replace A with A j +1 and by induction on n − j we have theresult. On the other hand, if S A j +1 , let S ′′ be the sector of A j +1 withsector germ S ′′ y = S y . Then ℓ ( d ( T ′ , S j +1 )) = ℓ ( d ( T, S j +1 )) − . S ′′ and T , together with A j +1 as our new A , by induction there exists an apartment A containing S ′′ y and T , with ℓ ( δ ( S ′′ y , T y )) ≤ ℓ ( d ( S ′′∞ , T ∞ ). However, S y = S ′′ y and ℓ ( d ( S ′′∞ , T ∞ )) ≤ ℓ ( d ( S ∞ , T ∞ )) − . Hence ℓ ( δ ( S y , T y )) ≤ ℓ ( d ( S, T ))as desired. Note that if equality holds, then in each case, the apartment A j contains S (where we take A j as the apartment containing S j and S y in theproof), and in particular, A n contains both S and T as desired. Corollary 6 If S and T are sectors of ∆ based at y , and δ ( S y , T y ) is maxi-mal, then S and T are contained in a common apartment. Proof:
Since δ ( S y , T y ) is maximal, Lemma 5 implies that ℓ ( δ ( S y , T y )) = ℓ ( d ( S, T )). However, in this case the lemma implies the existence of anapartment A containing S and T .From here we can now prove the first of our two theorems. Theorem 7
Suppose ∆ satisfies (A1), (A2), (A3), (A4), and (SE). Let S and T be sectors of ∆ based at x and y respectively. Then there exists anapartment A of ∆ containing S x and T y . That is, two sector germs arecontained in a common apartment. Proof:
We begin by showing that S x and y are contained in a commonapartment B . By (A2), there exists an apartment A ′ containing x and y .Let T ′ be a sector of A ′ based at x containing y . By Lemma 5 there exists anapartment B of ∆ containing S x and T ′ . Hence B contains S x and y . Take S ′ to be the sector of B based at y containing S x . Again by Lemma 5, thereexists an apartment A containing T y and S ′ . Since S x ⊂ S ′ , it follows that A contains T y and S x as desired.We now wish to show that condition (SE) can replace conditions (A5)and (A6) in the definition of an affine Λ-building. Since we have alreadyshown that (SE) is satisfied by an affine Λ-building ∆, and that (A6) canbe replaced by (SE), it remains to show that (A1)–(A4) and (SE) togetherimply the sector-retraction condition (A5). Our argument will proceed alongthe following lines. First we use Lemma 5 to define the retraction ρ S x ,A foran apartment A and a sector-germ S x ⊂ A and show that this retraction9reserves distances on sets X contained in a common apartment with S x .We then will show that (SE) implies a slightly stronger exchange condition,namely that if S x is a sector germ having a sector-panel germ in an apartment A , then there exist apartments A ′ and A ′′ such that S x ⊂ A ′ ∩ A ′′ and A ⊆ A ′ ∪ A ′′ . Given this condition, we will then be able to show that givenan apartment B , a point y ∈ B , and a sector-germ S x , there exists a sector T containing y based at x such that B contains a subsector of T . Since anytwo sectors based at x that contain a common subsector must be equal, itfollows that A is contained in a finite union of closed convex subsets X , . . . , X n , each of which are contained in an apartment with S x . Consequently, theretraction ρ S x ,A preserves distances on each of the X i s. Given any pair ofpoints y and z of B , it follows that we can find points y = y , y , . . . , y t = z such that y i − , y i ∈ X j i and y i is in the convex hull of y i − and y i +1 . As aresult, ρ S x ,A will preserve the distance between consecutive y i s, and using thetriangle inequality on A , it will follow that d ( ρ S x ,A ( y ) , ρ S x ,A ( z )) ≤ n X i =1 d ( ρ S x ,A ( y i − ) , ρ S x ,A ( y i ))= n X i =1 d ( y i − , y i ) = d ( y, z )Let A be an apartment of ∆, x ∈ A , and S a sector of A based at x .Let S x be the sector germ of S . We define ρ S x ,A as follows: For y ∈ ∆,Theorem 7 there exists an apartment B containing y and S x . Let f ∈ F be such that f (Σ) = B , g ∈ F be such that g (Σ) = A . Since S x ∈ A ∩ B ,by (A2) there exists w ∈ W such that g ( w ( f − ( S x ))) = S x . Moreover, as theidentity is the only element of W fixing a sector-germ, w is unique. Define ρ S x ,A ( y ) = g ( w ( f − ( y ))). By the compatibility condition, if B ′ is anotherapartment of ∆ containing S x and y with f ′ (Σ) = B ′ , it follows that thereexists w ′ ∈ W with g ( w ′ ( f ′− ( S x ))) = S x . This implies that w ′ ( f ′− ( S x )) = w ( f − ( S x )) . There also exists an element w ′′ ∈ W such that w ′′ ( f ′− ( S x )) = f − ( S x ).Sine the stabilizer of a sector germ is trivial, w ′′ = w − w ′ . But w ′′ ( f ′− ( y )) =10 − ( y ) by the compatibility condition (A2). Therefore, w ′ f ′− ( y ) = wf − ( y ),implying g ( w ′ f ′− ( y )) = g ( wf − ( y )), and thus ρ S x ,A is well-defined. Sincethe map w ∈ W preserves the distance on Σ, it follows that d ( y, z ) = d ( ρ S x ,A ( y ) , ρ S x ,A ( z )) for all y, z ∈ ∆ such that y , z , and S x are containedin a common apartment.We next show an even stronger exchange condition is satisfied. Lemma 8
Suppose ∆ satisfies conditions (A1), (A2), (A3), (A4), and (SE).Let A be an apartment of ∆ and S x be a sector germ of ∆ such that S x ∩ A is a sector-panel germ P x . Then there exists apartments A ′ and A ′′ such that S x ∈ A ′ ∩ A ′′ and A ⊂ A ′ ∪ A ′′ . Proof:
By the strong exchange condition (SE) it suffices to show that thereis a sector S ′ of ∆ intersecting A in a sector-panel such that S ′ x = S x . Let T be a sector of A based at x having a sector-panel containing the sectorpanel germ of S x . (That is, if P ′ is the sector-panel of A having sector panelgerm P x , take T to be a sector having a sector-panel P ′ .) By Lemma 5there exists an apartment B containing T and S x . Let S ′ be the sector of B having sector-panel germ S x . Then S ′ has sector-panel P ′ . Moreover, byconvexity, if S ′ ∩ A = P ′ , then S x = S ′ x ⊂ A contrary to our hypothesis.Therefore S ′ ∩ A = P ′ and by (SE) there exists apartments A ′ and A ′′ suchthat S ′ ⊂ A ′ ∩ A ′′ and A ⊂ A ′ ∪ A ′′ .This exchange condition allows us to work with sector-germs based ata common point, much as in the simplicial buildings case one works withchambers in a spherical residue. Proposition 9
Suppose ∆ satisfies conditions (A1), (A2), (A3), (A4), and(SE). Let S x be a sector germ contained in an apartment A of ∆ , and B beanother apartment of ∆ . Then for every point y ∈ B there exists a sector T of B such that1. There exists a sector T ′ based at x parallel to T containing y , and2. There exists an apartment A ′ of ∆ containing T and S x . Proof:
By Theorem 7, for every sector T of B based at y , there existsan apartment B ′ of ∆ containing T y and S x let S ′ denote the sector of B ′ based at y containing S x . For y ∈ B , choose T such that ℓ ( δ ( T y , S ′ y )) ismaximal. If T y and S ′ y are not opposite (that is δ ( T y , S ′ y ) is not the longest11lement of W ) then let P y be a sector-panel germ of T y such that the wall M of B ′ through P y does not separate T y and S ′ . In the apartment B thereexists a sector R such that R y shares P y with T y . Moreover, since T y and S ′ lie on the same side of M , by convexity the apartment B ′′ containing S ′ and R y guaranteed by Lemma 8 also contains T y . In B ′′ , we then have ℓ ( δ ( R y , S ′ y )) = ℓ ( δ ( T y , S ′ y ) + 1, contradicting the choice of T . Hence wemay assume that T y and S ′ y are opposite. By Corollary 6, there exists anapartment A ′ of ∆ containing S ′ and T . But S x ⊂ S ′ , so that S x ⊂ A ′ .Moreover, since A ′ contains T , take T ′ be the sector based at x parallel to T (in A ′ ). Since T and S ′ were opposite sectors and x ∈ T , it follows that y ∈ T ′ , completing the proof of the proposition.Note that by convexity, there is at most one sector of ∆ parallel to T based at a point x . Consequently, Proposition 9 implies that there existfinitely many sectors S , S , . . . , S n of ∆ each based at x , such that S x iscontained in a common apartment A i with each of the S i . Corollary 10
Suppose ∆ satisfies conditions (A1), (A2), (A3), (A4), and(SE). Let S x be a sector germ contained in an apartment A of ∆ and B bean apartment of ∆ . Then there exists closed convex sets X , . . . , X n of B such that1. B = X ∪ · · · ∪ X n and2. Each X i lies in a common apartment with S x . Proof:
Let S , . . . , S n be the sectors from the above paragraph, and set X i = S i ∩ B .We are now ready to prove the map ρ S x ,A is a retraction. Proposition 11
Suppose ∆ satisfies conditions (A1), (A2), (A3), (A4), and(SE). Let S x be a sector germ contained in an apartment A of ∆ . Then themap ρ S x ,A is a retraction of ∆ onto A . Let ρ = ρ S x ,A . We have already shown that ρ : ∆ → A is well-defined, and ρ | A is the identity by definition. It remains to see that ρ diminishes distance.That is, to show for all y, z ∈ ∆ that d ( ρ ( y ) , ρ ( z )) ≤ d ( y, z ) . By definition, if there is an apartment B of ∆ containing S x , y , and z ,then d ( y, z ) = d ( ρ ( y ) , ρ ( z )), so the result holds true. Now, suppose y and z B containing y and z . ByCorollary 10 there exists closed convex sets X , . . . , X n such that B = ∪ ni =1 X i and each X i is contained in a common apartment with S x . Since each X i isconvex and closed, there exists a sequence of points y = y , y , . . . , y k = z such that y i − , y i ∈ X j i for some j , . . . , j k and y i is in the convex hull of y i − and y i +1 for i = 1 , . . . , k −
1. Then d ( y, z ) = k X i =1 d ( y i − , y i )= k X i =1 d ( ρ ( y i − ) , ρ ( y i )) ≥ d ( ρ ( y ) , ρ ( y k ))= d ( ρ ( y ) , ρ ( z ) , where we use the triangle inequality for d restricted to Σ in the next to thelast step. Thus, ρ is distance diminishing and hence a retraction with therequired properties.Summarizing, we have proven Theorem 12
Suppose (∆ , F ) satisfies conditions (A1), (A2), (A3), and(A4). Then conditions (A5) and (A6) together are equivalent to condition(SE). In particular, if (∆ , F ) satisfies conditions (A1), (A2), (A3), (A4),and (SE) then (∆ , F ) is an affine Λ -building. . References [B1] Bennett, Curtis, Affine Λ-buildings, Ph.D. Thesis, University ofChicago, 1990.[B2] Bennett, Curtis, Affine Λ-buildings I,
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