DOI 10.1515 / ADVGEOM.2009.025 de Gruyter 2009
Convex subspace closure of the point shadow of an apartment of a spherical building
Anna Kasikova∗
(Communicated by A. Pasini)
Abstract. LetΣbe the point-line truncation of an augmentedJ-Grassmann geometry of a spher- ical buildingB. We show that if|J|=1, or if|J| ≥2 and some additional conditions hold, then the convex subspace closure inΣof the point shadow of an apartment ofBis the entire spaceΣ.
1 Introduction
LetBbe a spherical building over a type setI. LetJ ⊆Iand letΣ = (P,L)be the point- line truncation of theJ-Grassmann geometry ofB. The geometryΣcan be regarded as a point-line space (see 12.15 of [11] or Lemma 2.4 of [7]). LetS ⊆ P. Thespanor the subspace closureof S inΣis the intersection of all subspaces ofΣcontainingS. The convex subspace closureof the setSinΣis the intersection of all convex subspaces ofΣ containingS.
When|J| = 1, the question whether or not the point shadow of an apartment ofB spans the spaceΣhas been investigated in [4] and [2]. The answer varies depending on the Dynkin diagram ofBand the choice of the setJ.
The main result of the present paper is Theorem 1.1 of Section 1.3. This theorem shows that, if the J-Grassmann geometry Σ has rank at least two, and |J| = 1, or
|J| ≥ 2 and some additional condition holds, then the convex subspace closure of the point shadow of an apartment ofB inΣis the entire spaceΣ. We chose to state The- orem 1.1 in terms of point-line truncations of augmentedJ-Grassmann geometries (de- fined in Section 1.3), instead ofJ-Grassmann geometries, so that we can avoid making exceptions for low-rank cases. If the rank of theJ-Grassmann geometry is at least two, then the point-line truncation of theJ-Grassmann geometry coincides with the point-line truncation of the augmentedJ-Grassmann geometry.
∗Most of this work was completed while the author was a Visiting Research Instructor at Michigan State University.
The structure of the paper is as follows. The remaining subsections of the present section contain most of the necessary definitions and statement of the main result (The- orem 1.1 of Section 1.3). In Section 2 we prove a property of buildings similar to the Deletion Condition for Coxeter groups (Proposition 2.1), and then state some corollaries of this property required in the proof of Theorem 1.1.
Section 3 contains some technical definitions used in the proof of Theorem 1.1, and Section 4 contains the proof of Theorem 1.1. In Section 6 we give two examples of ap- plications of Theorem 1.1. Section 5 contains two results regarding the convex closure of a set of chambers in a building, related to the caseSp =∅of Theorem 1.1 (see Proposi- tions 5.2 and 5.4).
1.1 Definitions of metric properties of graphs and chamber systems. SupposeG= (V, E)is a graph. A walk of lengthninGis a sequence of vertices(x0, . . . , xn)such that{xi, xi+1} ∈Efor alli∈ {0, . . . , n−1}. The length of the walkwis denotedl(w).
Ifw1= (x0, . . . , xk)andw2= (xk, . . . , xn)are walks inG, then we denotew1◦w2the walk(x0, . . . , xk, xk+1, . . . , xn); the walkw1◦w2is called theconcatenationofw1and w2. IfGis the graph of a chamber system then walks inGare also calledgalleries.
AgeodesicfromxtoyinGis a walk fromxtoyof the smallest possible length. The distance between two verticesx, y∈V is the length of a geodesic fromxtoyinG. The distance fromxtoyinGwill be denoteddG(x, y)or justd(x, y).
A subsetXofV isconvexinGif, for everyx, y∈X, the vertices of every geodesic fromxtoyinGlie inX. A subgraph ofGis convex inGif its set of vertices is convex inG.
LetG = (V, E)be a graph, and suppose thatG0 = (V0, E0) is a subgraph ofG.
We say thatG0 isstrongly gatedinGif, for every x ∈ V, there is a vertexx0 ∈ V0 such that, for everyy ∈ V0, we have dG(x, y) = dG(x, x0) + dG0(x0, y). The vertex x0 is called thegateofxinG0 and is denotedgateG0(x). For a subsetX ofV we let GateG0(X) ={y∈V0|y= gateG0(x), x∈X}.
SupposeCis a chamber system over the type setI, such that every edge is labelled by a one-element subset ofI. Letw = (c0, . . . , cn)be a gallery inCand suppose that, for everyi∈ {1, . . . , n}, the label of the edge{ci−1, ci}is{ti},ti∈I. Then we say that the typeof the gallerywist(w) =t1. . . tn, a word in the free monoid of words onI.
SupposeCis a chamber system over a type setIwith graph(C, E)and labelling map λ. Suppose G = (C0, E0) is a subgraph of(C, E). We denote typ(G) the set of all elements ofIthat appear in labels of edges ofG, that istyp(G) =S
e∈E0λ(e). IfXis a set of chambers ofC, thentyp(X)denotestyp(G), whereGis the subgraph ofCinduced onX. Most of the time we will use the same letter to denote a set of chambers ofC, and the chamber subsystem and the subgraph induced on this set inC.
1.2 Buildings and properties of buildings. LetIbe a nonempty set and letM be a Coxeter matrix overI. That is,M is a map fromI×IintoN∪ {∞}that takes(i, j)to mij, such thatmii=1 for alli∈I, andmij ≥2 for alli, j∈Iwithi6=j. Achamber system of typeM overIis a connected chamber system in which, for every{i, j} ⊆ I, all residues of type{i, j}are chamber systems of generalizedmij-gons. Note that, in a
chamber system of typeM, every edge is labelled by a one-element subset ofI, and every rank one residue contains at least two chambers.
LetM be a Coxeter matrix over a setIand letW be the Coxeter group of typeM with the set of generatorsS ={si |i ∈I}. Abuildingof typeM overIis a chamber systemBof typeMthat satisfies one of the following three equivalent conditions.
(P) If two galleriesw andw0 of reduced typest1. . . tn andt01. . . t0n0 have the same initial and terminal chambers, thenst1. . . stn=st01. . . st0
n0 inW.
(G) Every gallery ofBof reduced type is a geodesic.
(RG) All residues ofBare strongly gated inB.
Conditions (P) and (G) refer to galleries of reduced type. These are defined as follows.
SupposeCis a chamber system of typeM overI. LetWbe a Coxeter group of typeM overI with generatorsS ={si | i ∈ I}. Letwbe a gallery inBof typet1. . . tn and letr∈ Wbe such thatr=st1. . . stn. Thenwis agallery of reduced typeifst1. . . stn is a word of the smallest possible length in the free monoid of words onS such that st1. . . stn =rinW.
SupposeBis a building of typeM over the type setI. LetW denote the Coxeter group of typeM and, also, the Coxeter chamber system of typeM. AnapartmentofBis any map fromWintoBwhich is an isomorphism of chamber systems overI. The image ofWunder this isomorphism is also called an apartment ofB(cf. Proposition 2.1 of [6]).
For ease of reference, below we list some properties of buildings. Their proofs can be found in [12], [9], and [10].
(B0) Every pair of chambers ofBis contained in some apartment ofB.
(B1) Apartments and residues are convex induced chamber subsystems ofB.
(B2) SupposeAandA0 are apartments ofBand supposeR1 andR2 are residues ofB such thatA∩Ri6=∅andA0∩Ri6=∅fori=1,2. Then there is an isomorphism of chamber systems overI,ϕ:A→A0, such that, fori=1 and 2,ϕ(A∩Ri) = A0∩Ri.
(B3) IfRis a residue ofBof typeJ, thenRis a building of typeM|J. (B4) If{Qα|α∈S}is a family of residues ofBandT
α∈SQα6=∅, thenT
α∈SQαis a residue ofBof typeT
α∈Styp(Qα).
(B5) SupposeR,P, andQare residues ofBsuch thatR∩P 6= ∅,R∩Q 6= ∅, and P∩Q6=∅. ThenR∩P∩Q6=∅.
(B6) SupposeRis a residue ofB, and supposeA1is an apartment ofR. Then there exists an apartmentAofBsuch thatA1 ⊆A.
A Coxeter diagramM overIissphericalif the Coxeter group of typeM is finite. A buildingBof typeM overIis spherical ifM is spherical. IfBis a spherical building, then all residues ofBare spherical buildings.
1.3 Point-line spaces associated with buildings and statement of the main result.
Anincidence geometry over a type setI is a multipartite graph with parts labelled by elements ofI. For everyi∈I, the vertices of the graph belonging to the part labelledi are called theobjectsof the geometry of typei. Two adjacent vertices of the graph are said to beincidentobjects of the geometry. IfΓ is an incidence geometry andO is an object ofΓ, then we denoteΓ(O)the set of all objects ofΓincident withO.
Apoint-line space(P,L)is a set of pointsP together with a setLof subsets ofP such that|L| ≥2 for everyL∈ L. An incidence geometry over the type set{point,line}
can be viewed as a point-line space, if every line of the geometry is incident with at least two points and, for every pair of distinct lines, the sets of points incident with them never coincide.
LetΣ = (P,L)be a point-line space. The point-collinearity graph ofΣis a graph (P,E)with vertex setP in which two vertices are adjacent if and only if there is a line containing both. A subsetSofP is asubspaceofΣif every line that meetsX in at least two distinct points is contained inX. A subsetSofPis aconvex subspaceofΣifSis a subspace ofΣandSis convex in(P,E).
LetΣ = (P,L)be a point-line space. SupposeSis a subset ofP. We denotehSiΣor hSithe intersection of all subspaces ofΣcontainingS, and we say thathSiis thespanor thesubspace closureofSinΣ. We denotehhSiiΣorhhSiithe intersection of all convex subspaces of(P,L)containingS, and we say thathhSiiis theconvex subspace closureof Sin(P,L).
LetBbe a building of typeM overI, and letDbe the diagram graph ofM andB.
That is,Dis a graph with vertex setIin which two distinct verticesi, j∈Iare adjacent if and only ifmij ≥3. We now define the augmentedJ-Grassmann geometry ofB.
LetSp⊆Iand letI00={k∈N|1≤k≤ |I0|}, whereI0is the union of vertex sets of all connected components ofDthat meetI−Sp. LetΓbe the(I−Sp)-Grassmann geometry ofBdefined as in [6, 7]. ThenΓis a geometry over the type setI00. For an objectOofΓwe denoteROthe residue ofBcorresponding toO.
Theaugmented(I−Sp)-Grassmann geometry ofBis a geometryΓ0over the type setI000=I00∪ {∞}, obtained fromΓby adding an objectOof type∞incident with all objects ofΓ. The residueROofBcorresponding to the objectOis the entire buildingB.
Both, the J-Grassmann geometry and the augmentedJ-Grassmann geometry, can be constructed for any chamber systemC, not necessarily a building. We just define the diagram graph ofC as a graph with vertex set I, the type set of C, in which two verticesi, j∈Iare adjacent if and only if at least one residue ofCof type{i, j}is not a generalized digon.
LetBbe a building of typeM overI with diagram graphD. LetΣ = (P,L)be the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofB. Then the set of pointsPis in bijective correspondence with the set of all residues ofBof typeSp, and the set of linesLis in bijective correspondence with the set of all residues ofBof all possible typesT, such thatT ={i} ∪(Sp−(D0,1(i)∩Sp))for somei∈I−Sp, (we denoteD0,1(i)the set of all vertices of the graphDat distance 0 or 1 fromi). A point p∈ Pand a lineL ∈ Lare incident inΣif and only ifRp∩RL 6= ∅. Every rank one residue ofBis contained in a residue corresponding to a point or a line ofΣ.
For every buildingBand every setSp, the set of points of the augmented(I−Sp)- Grassmann geometryΓ0is nonempty, and the point-collinearity graph ofΓ0is connected.
This makes the augmentedJ-Grassmann geometry more convenient for the purposes of the present paper than theJ-Grassmann geometry. If|I0| ≥2, then the point-line trunca- tion of the augmented(I−Sp)-Grassmann geometry ofBcoincides with the point-line truncation of the(I−Sp)-Grassmann geometry ofB.
LetBbe a building of typeM overIwith diagram graphD, and suppose thatΣ = (P,L)is the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofB.
LetX be a set of chambers ofB. We denotePX the set of all pointsp∈ P such that Rp∩X 6=∅, and we say thatPXis thepoint shadowofX.
Our goal in the present paper is to prove the following theorem. For the definition of the opposition involution see Section 2.2 (or see [9]).
Theorem 1.1. LetBbe a spherical building with Coxeter diagramMoverI. LetSp⊆I and letΣ = (P,L)be the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofB. Assume that at least one of the following two conditions holds.
(C1) |I−Sp|=1.
(C2) The setI−Spis stabilized by the automorphism of the diagramM induced by the opposition involution of the Coxeter chamber system of typeM (this includes the caseSp=I).
LetAbe an apartment ofB. ThenhhPAii=P.
If the buildingBis not assumed to be spherical then the conclusion of Theorem 1.1 fails. This can be seen in the following example. LetB be a generalizedm-gon with m=∞over the type setI={i, j}, and letSp=∅. LetΣbe the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofB, and letGdenote the point-collinearity graph ofΣ. The points ofΣare the chambers ofB, the lines ofΣare the panels ofB, and the graphGis the graph of the chamber systemB. LetAbe an apartment ofB. Then Ais a doubly infinite gallery with edges labelled alternately by{i}and{j}, consisting of pairwise distinct chambers. We havePA = A, andhhPAiiis the union of the sets of chambers of the panels ofBthat meetA. Therefore, ifBis not thin, thenhhPAiiis not the entire buildingBin this case.
No example of a spherical building, satisfying at least one of the conditions (C1) and (C2) such that the conclusion of Theorem 1.1 fails, is known to the author.
When the diagram graph of the building has more than one connected component, Theorem 1.1 can be combined with Proposition 1.2 below. Proposition 3.2, referred to in the proof of Proposition 1.2, is just Proposition 3.6 of [7] restated for augmented Grass- mann geometries (see Section 3).
Proposition 1.2. LetBbe a building with Coxeter diagramMoverIand diagram graph D. LetSp⊆Iand letΣ = (P,L)be the point-line truncation of the augmented(I−Sp)- Grassmann geometry ofB.
LetI1, . . . , Ik be the vertex sets of the connected components of the diagram graph D, where kis a positive integer. For every i ∈ {1, . . . , k}, for every residueRi ofB of typeIi, and for every apartmentAi ofRi, suppose that hh(Pi)AiiiΣi = Pi, where Σi = (Pi,Li)is the point-line truncation of the(Ii−(Ii∩Sp))-Grassmann geometry ofRi.
LetAbe an apartment ofB. ThenhhPAii=P.
Proof. LetAbe an apartment ofB. Letcbe a chamber ofA. For everyi∈ {1, . . . , k}, letAibe the residue ofAof typeIicontainingc. The chamber systemAis the product of the chamber systemsAi.
For everyi∈ {1, . . . , k}, letRibe the residue ofBof typeIicontainingAi, and let Σi = (Pi,Li)be the point-line truncation of the(Ii−(Ii∩Sp))-Grassmann geometry ofRi.
Letϕi : Pi → P be the map that takes a point or lineO ∈ Pi∪ Li to the point or lineO0 ∈ P ∪ L such thatRO0∩Ri is the residue of Ri corresponding toO. By Proposition 3.2ϕiis an isomorphism ofΣiontoΣ|PRi, whereΣ|PRi = (PRi,L|PRi) andL|PRiis the set of lines ofΣmeetingPRiin at least two distinct points.
By Theorem 6.1 of [6] PRi is a convex subspace ofΣ, therefore the convex sub- space closure inΣ|PRi coincides with the convex subspace closure insideΣ. Since by hypothesishh(Pi)AiiiΣi =Pi, we obtain thathhPAiii=PRi.
The above shows that, for everyi ∈ {1, . . . , k},PRi ⊆ hhPAii. We claim that this implies thathhPAii=P. It suffices to consider the casek=2 since the other cases follow from this case by induction onk.
Supposek=2. ThenΣis the product geometryΣ = Σ|PR1×Σ|PR2 ∼= Σ1×Σ2. Letp∈ P. LetQ1andQ2be the residues ofBof typesI1andI2intersectingRp, and let Σ01 = Σ|PQ1andΣ02= Σ|PQ2. We haveΣ01∼= Σ1,Σ02∼= Σ2, andΣ = Σ|PQ1×Σ|PQ2 ∼= Σ01×Σ02.
Letp1andp2be the projections ofpontoPR1 andPR2. That is,{p1}=PR1∩ PQ2 and{p2}=PR2∩ PQ1. Any geodesic fromp1top2inGprojects onto a walk fromp1top inPQ2and onto a walk fromptop2inPQ1. For every edge ofG, one projection is an edge and the other projection is a single vertex. ThereforedG(p1, p2)≥dG(p1, p) + dG(p, p2).
This shows thatplies on a geodesic fromp1 top2inG. Sincep1, p2 ∈ hhPAii, we have p∈ hhPAii.
We showed that every point ofP is inhhPAii. Therefore,hhPAii=P. 2
2 A property of buildings and its corollaries
The main purpose of this section is to prove Proposition 2.6 of Section 2.2, that will be used in the proof of Theorem 1.1. The proof of Proposition 2.6 depends on Proposition 2.1 to construct a geodesic of a certain type between two chambers of a buildingB. If the buildingBis thin, that is ifBis a Coxeter chamber system, then Proposition 2.1 becomes the Deletion Condition for Coxeter groups (the statement of the Deletion Condition for Coxeter groups can be found in [5]).
2.1 A property of buildings. The proof of Proposition 2.1 uses the following property of Coxeter chamber systems. SupposeCis a Coxeter chamber system over a type setI.
Letxandy be chambers ofCand letW be a geodesic fromxtoyinC. Supposeris a reflection in the Coxeter group ofC. That is,ris an automorphism of the chamber system Cthat interchanges two vertices of some edge ofC. Then there are two possibilities: (1) rstabilizes no edge ofW or (2) rstabilizes exactly one edge ofW. In Case (1) the reflectionrdoes not stabilize any edge of any geodesic fromxtoyinC, and in Case (2) the reflectionrstabilizes exactly one edge of every geodesic fromxtoyinC. For a proof of this property of Coxeter groups see [9].
Proposition 2.1. LetBbe a building of typeMover a type setI. Letxandybe chambers ofB, and letW be a gallery fromxtoy inB. Suppose that the type ofW ist1. . . tk, wheret1, . . . , tk∈I. IfW is not a geodesic fromxtoyinB, then there is a galleryW0 fromxtoyof typet1. . .tbj. . . tk, or of typet1. . .tbi. . .tbj. . . tk, wherei, j∈ {1, . . . , k}
andtbαindicates that the elementtαis omitted.
Proof. Suppose thatW = (c0, . . . , ck)is a gallery fromc0 =xtock =y inBof type t1. . . tk, and suppose thatW is not a geodesic. Letcj+1be the first chamber ofW such thatd(x, cj+1)6= d(x, cj) +1. Note that 1≤j≤k−1 andcj6=x, y.
LetW1be the part of the galleryW beginning withxand ending withcj, and letW2 be the part ofWbeginning withcj+1and ending withy. ThenW =W1◦(cj, cj+1)◦W2, and by the choice ofcj+1the galleryW1is a geodesic fromxtocj.
Case1.Suppose first thatd(x, cj+1) = d(x, cj). LetQbe the panel ofBof type{tj} containingcjandcj+1. LetAj be an apartment ofBcontainingxandcj, letAj+1be an apartment ofBcontainingxandcj+1, and letg = gateQ(x). ApartmentsAjandAj+1 exist by (B0).
Since by (B1)AjandAj+1are convex inB, we haveAj∩Q={cj, g}andAj+1∩ Q={cj+1, g}. By (B2) there is an isomorphismϕ:Aj → Aj+1such thatϕ(x) =x, ϕ(g) =g, andϕ(cj) =cj+1.
By convexity of Aj (property (B1)), all vertices of W1 lie in Aj. Let W10 be the gallery which is the image ofW1underϕ. ThenW10◦W2 is a gallery fromxtoyinB.
The type ofW10ist1. . . tj−1and the type ofW2 istj+1. . . tk, thereforet(W10◦W2) = t1. . . tj−1tj+1. . . tk.
Case2.Suppose thatd(x, cj+1) = d(x, cj)−1. LetAbe an apartment ofBcontaining xandcj. LetW10be a geodesic fromxtocj+1inB. The galleriesW1andW10◦(cj+1, cj) are both geodesics fromxtocj inB. SinceAis convex inB, andx, cj ∈ A, all the chambers of bothW1andW10◦(cj+1, cj)lie inA.
Letrbe the reflection in the Coxeter group ofA such thatr(cj) = cj+1. Sincer stabilizes the edge{cj, cj+1}ofW10◦(cj+1, cj), it must stabilize exactly one edge ofW1. Letciandci+1be the chambers ofW1such thatr(ci) =ci+1. LetU be the part of W1beginning withxand ending withci, and letV be the part ofW1beginning withci+1 and ending withcj. ThenW1 =U◦(ci, ci+1)◦V.
LetV0 be the image ofV underr. ThenV0 begins withci and ends withcj+1. Let W0 =U ◦V0◦W2. ThenW0is a gallery fromxtoyinB. We havet(V0) = t(V)and t(W0) = t(U) t(V0) t(W2). Thereforet(W0) =t1. . . ti−1ti+1. . . tj−1tj+1. . . tk. 2 The proof of Case 2 in the above proposition is essentially a proof of the Deletion Condition for Coxeter groups (cf. Exercise 4, p. 24, of [9]) and can be replaced with a reference to the Exchange Condition or the Deletion Condition.
The following corollary is immediate from Proposition 2.1.
Corollary 2.2. LetBbe a building of typeMover a type setI. Letxandybe chambers ofB, and supposeW is a gallery fromxtoyinB. IfW is not a geodesic, then there is a geodesicW0fromxtoyinBsuch thatt(W0)is obtained fromt(W)by omitting one or more terms.
2.2 Geodesics between two points in (P,L). In this section we are going to use Proposition 2.1, together with Lemma 2.5, to show existence of certain geodesics in the point-collinearity graph of the augmentedJ-Grassmann geometry of a spherical building.
The result is stated as Proposition 2.6.
First, we prove Proposition 2.3 and Corollary 2.4, that apply to all buildings, not necessarily spherical. Corollary 2.4 shows that, ifCis a convex induced chamber sub- system of a buildingB, andΣis the(I−Sp)-Grassmann geometry ofB, then the point- collinearity graph of theJ-Grassmann geometry ofC, corresponding toΣ, embeds into the point-collinearity graph ofΣisometrically.
We need the following notation. LetBbe a building of typeM overI. LetSp ⊆I and letΣ = (P,L)be the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofB. LetGdenote the point-collinearity graph ofΣ.
Since every residue ofBof rank 1 is contained in a residue corresponding to a point or line ofΣ, every galleryW = (c0, . . . , cn)ofBdetermines a walkw = (p0, . . . , pk) in the graphG, that consists of the points corresponding to the residues ofBof typeSp traversed by the galleryW. We denote this walkwG(W), and we assume that no two consecutive points ofwG(W)are equal to each other.
Conversely, supposew= (p0, . . . , pn)is a walk inG. Letx∈Rp0and lety∈Rpn. Every point residue is a connected chamber system. Also, by Lemma 3.10 of [7], ifhp, qi is a line ofΣ, then there is a chamber inRp connected by an edge to a chamber inRq. Combining these two facts we obtain that there is a galleryW fromxtoyinBsuch that wG(W) =w.
Proposition 2.3. LetBbe a building of typeMoverI. LetSp⊆Iand letΣ = (P,L)be the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofB. Suppose p, q∈ P. Letx∈Rpand lety ∈Rq. Then there is a geodesicW fromxtoyinB, such thatwG(W)is a geodesic fromptoqinG.
Proof. Letw = (p0, . . . , pn)be a geodesic fromp = p0 toq =pn inG. LetW be a gallery inBbeginning withxand ending withy, such thatwG(W) =w.
Suppose thatW is not a geodesic inB. By Corollary 2.2, there is a geodesicW0from xtoyinB, such thatt(W0)can be obtained fromt(W)by omitting one or more terms.
Letw0 = wG(W0). Thenl(w0)≤l(w). Thereforew0is a geodesic fromptoqinG. 2 The following is an immediate corollary of Proposition 2.3.
Corollary 2.4. LetBbe a building of typeM overI. LetSp ⊆Iand letΣ = (P,L) be the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofB. LetG denote the point-collinearity graph ofΣ.
SupposeCis a convex set of chambers inB, and letCbe the chamber subsystem of Binduced onC. LetΣ(C) = (P(C),L(C))be the point-line truncation of the augmented (T −(T ∩Sp))-Grassmann geometry ofC, whereT = typ(C). LetG(C)denote the point-collinearity graph ofΣ(C).
Letϕbe the map that takes a pointp∈ PCto the pointp0 ∈ P(C)corresponding to the residueRp∩CofC. Then, for allp, q∈ PCwe havedG(p, q) = dG(C)(ϕ(p), ϕ(q)).
LetBbe a spherical building of typeM overI. Then the graph of the chamber system Bhas finite diameter. Letddenote the diameter ofB. Two chamberscandc0 ofBare oppositeinBifd(c, c0) =d.
LetCbe a Coxeter chamber system of typeM. Then the diameter ofCisd, and every chamber ofChas a unique opposite inC. The mapopC :C → Cthat takes every chamber ofCto its opposite inCis, in general, a non-type preserving automorphism of the chamber systemC. That is,opC is an automorphism of the graph of the chamber systemCwhich, for everyi ∈ I, maps the edges ofClabelled{i} to the edges ofC labelled{opM(i)}, whereopM denotes the automorphism of the diagramM induced byopC. The mapopC is called theopposition involutionofC. SupposeQis a residue ofC. Then the image of QunderopCis a residue ofCand is called theoppositeresidue ofQinC.
SupposeQandQ0are residues of the buildingB. ThenQandQ0areoppositeinBif there is an apartmentAofBsuch thatQ∩AandQ0∩Aare opposite inA. IfQandQ0 are opposite residues ofB, then they are opposite in every apartment ofBmeeting both.
These definitions and proofs of the facts used in them can be found in [11] or [9].
LetB be a spherical building and let ddenote the diameter ofB. SupposeQ and Q0 are residues of B. ThenQ andQ0 are opposite in Bif and only if the map of Q intoQ0 defined byx 7→ gateQ0(x)is a bijection ofQontoQ0, and, for everyc ∈ Q, dB(c,gateQ0(c)) =d−dQ0, wheredQ0 denotes the diameter of the residueQ0.
Supposecandc0are opposite chambers ofB, and supposeQis a residue ofBcon- tainingc. Then there is a unique residueQ0 ofBthat containsc0 and is opposite toQ inB.
Lemma 2.5. LetCbe a spherical building of typeM overI. LetSp ⊆Iand letΣ = (P,L)be the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofC.
LetGbe the point-collinearity graph ofΣ.
Suppose thatC is thin, and suppose that the setI−Sp is stabilized byopM, the automorphism ofM induced by the opposition involution ofC.
Letp, q ∈ P be such that the residuesRpandRqare opposite inC. Then, for every α∈I−Sp, there is a geodesicw = (p0, p1, . . . , pn)fromp=p0toq=pninG, such thatRpandRp1are connected by an edge labelled{α}.
Proof. LetopCbe the opposition involution ofC. By the hypothesis the setSpis stabilized byopM, therefore the image underopC of a residue ofCof typeSpis a residue ofCof typeSp. In particular, there exist pointspandqas in the hypothesis.
We can assume thatI−Sp 6=∅. Therefore,p6=q. Letx∈Rpand letz= opC(x).
Thenz∈Rq. By Proposition 2.3 there is a geodesicUfromxtozinC, such thatwG(U) is a geodesic fromptoqinG. Letu= wG(U)and suppose thatu= (p0, p1, . . . , pn), wherep0 =pandpn =q.
LetV be the image ofU underopC. Sincexandzare opposite inC, the galleryV is a gallery fromztox.
Letv= wG(V)and suppose thatv= (q0, . . . , qm). By hypothesisopM stabilizes the setSp. Thereforeq0=qandqm=p. Furthermore,m=nand, for everyi∈ {0, . . . , n}, Rqiis the residue ofCopposite toRpi.
Fori∈ {0, . . . , n}, letϕibe an automorphism ofCtakingRpitoRp. The automor- phismϕiexists since the automorphism group ofCis transitive on the set of chambers of C. Letψibe the automorphism ofGinduced byϕi.
Letui= (pi, pi+1, . . . , q, q1, . . . , qi). SinceRqiis the unique residue ofCopposite to Rpi, andRqis the unique residue ofCopposite toRp, andψi(pi) =p, we haveψi(qi) = q. Thereforeψimaps the walkuito a walku0i fromptoq. Sincel(ui) = l(u) =n, the walku0iis a geodesic fromptoqinG.
Letα∈I−Sp. Since the chambersxandzare opposite inC, the type of the gallery Urepresents the longest word in the Coxeter group corresponding toC. This implies that every element ofIoccurs as the label of an edge ofU at least once. Therefore, there is i∈ {0, . . . , n−1}such thatRpiandRpi+1are connected inU by an edge labelled{α}.
Then the walku0i, constructed in the preceding paragraph, is a geodesic fromptoq, and RpandRψi(pi+1)are connected by an edge labelled{α}. 2
Proposition 2.6. LetBbe a spherical building of typeM over I. Let Sp ⊆ I and let Σ = (P,L)be the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofB. LetGbe the point-collinearity graph ofΣ.
Letp, q∈ Pand supposep6=q. Assume that at least one of the following conditions holds.
(1) |I−Sp|=1.
(2) The setI−Spis stabilized byopM, and the residuesRpandRqare opposite inB.
Then, for everyα∈I−Sp, there is a geodesicw= (p0, p1, . . . , pn)fromp=p0to q=pninG, such thatRpandRp1are connected by an edge labelled{α}.
Proof. Suppose Part (1) of the hypothesis holds. Letα∈ I−Sp, and denote byw = (p0, p1, . . . , pn)a geodesic fromp0 =ptopn = qinG. By hypothesis|I−Sp| = 1, thereforeI−Sp={α}. Sincepandp1are distinct collinear points ofΣ, by Lemma 3.10 of [7] the residuesRpandRp1are connected by an edge, and the label of this edge is not inSp. Therefore, the edge is labelled{α}, andwis the required geodesic.
Suppose Part (2) of the hypothesis holds. Letα∈I−Sp. LetAbe an apartment ofB that intersects bothRpandRq. LetΣ(A) = (P(A),L(A))be the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofA, and letG(A)be the point-collinearity graph ofΣ(A).
Letϕbe the map that takes a point or lineO ofΣ(A)to the point or lineO0 ofΣ such thatRO0∩Ais the residue ofAcorresponding toO. By Proposition 3.6 of [7] (see Section 3, Proposition 3.2) the mapϕexists and is an isomorphism ofΣ(A)intoΣthat mapsP(A)ontoPA.
We have p, q ∈ PA. Letv be a geodesic from ϕ−1(p) toϕ−1(q)in G(A). Let w= (p0, . . . , pn)be the image ofvunderϕ. We havep0=p,pn=q. SinceAis convex inB, by Corollary 2.4wis a geodesic fromptoqinG. By Lemma 2.5 we can choose the geodesicvso thatRpandRp1are connected by an edge labelled{α}. 2
3 Additional definitions
LetM be a Coxeter matrix over a setI, and letBbe a building of typeM. We denote byDthe diagram graph ofM andB. LetSp ⊆Iand letΣ = (P,L)be the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofB.
First, we recall some notation introduced in [6]. LetT ⊆I. ThenKSp(T)denotes the union of the vertex sets of all connected components of the graphD|T that meetI−Sp. IfQis a residue ofBof typeT, andQ0is a residue ofQof typeKSp(T), thenPQ=PQ0. SupposeCis a chamber subsystem of the buildingB, and suppose that the type set of CisT. Then we denoteΣ(C) = (P(C),L(C))the point-line truncation of the augmented (T −(T ∩Sp))-Grassmann geometry ofC(this differs from the notation in [7], where Σ(C)denotes the point-line truncation of(T−(T∩Sp))-Grassmann geometry ofC).
Suppose thatX is a subset ofP. We denote byL|Xthe set of all lines ofΣthat meet X in at least two distinct points. We letΣ|X be the point-line geometry whose set of points isX, whose set of lines isL|X, and whose incidence is inherited fromΣ.
LetchamB:2P →2Bbe the map defined by the rule that, ifX is a subset ofP, then chamB(X) = S
p∈XRp. The setchamB(X)can be viewed as the inverse image ofX under the map 2B →2P that takes a set of chambersY ofBto its point shadowPY. For a subsetP0ofP, we havePchamB(P0)=P0.
LetX be a subset ofP. We denote by Xe the set of pointsS
L∈L|XPL, wherePL
denotes the set of all points ofΓincident with the lineL. That is,Xeconsists of all points ofΓincident inΓwith lines ofΣ|X. We haveX ⊆X.e
SupposeCis a set of chambers ofB. Then we letCedenote the setchamB(PfC). We havePC⊆PfCandC⊆C. Ife Cis an apartment ofB, then Lemma 3.1 below shows that Cecontains every panel ofBon every chamber ofC.
Lemma 3.1. LetBbe a building of typeM overI, letSp⊆I, and letΣ = (P,L)be the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofB.
SupposeAis an apartment ofB. Then, for every chamberc∈A, the setAecontains all panels ofBonc.
Proof. Letc∈A, leti∈I, and letQbe a panel ofBoncof type{i}. Suppose first that i∈Sp. Letp∈ PAbe such thatc∈Rp. ThenQ⊆RpandRp⊆A, thereforee Q⊆A.e
Supposei6∈Sp. LetT ={i} ∪(Sp−(D0,1(i)∩Sp)). Then residues of typeT of Bcorrespond to lines ofΣ. LetL ∈ Lbe the line corresponding to the residue of type T containing Q, and letPL denote the set of all points of Σincident withL. A point q∈ Pis incident inΣwith the lineLif and only ifRq∩Q6=∅. ThereforePL =PQ. Moreover, ifq∈ PL, then by (B4)Rq∩Qis a single chamber.
SinceQ∩Ais a panel ofA, the lineLis incident with two distinct points ofPA. ThereforeL∈ PAandPL⊆PfA. This impliesQ⊆chamB(PL)⊆A.e 2 We close this section with the following observation. Let B be a building over a type setI. SupposeSp ⊆ I is such that |KSp(I)| = 1. Then the set of lines of the
(I−Sp)-Grassmann geometry ofBis empty but the set of lines of the augmented(I−Sp)- Grassmann geometry ofBis nonempty (it consists of exactly one line, whose correspond- ing residue is the buildingBitself). Similarly, ifSp=I(that is|KSp(I)|=0), then the set of points of the(I−Sp)-Grassmann geometry ofBis empty but the set of points of the augmented(I−Sp)-Grassmann geometry ofBis nonempty and consists of exactly one point, corresponding to the buildingB.
The absence of points or lines in the(T−(T∩Sp))-Grassmann geometry of the cham- ber subsystemCwhen|KSp(T)| ≤1 was the only reason why we required|KSp(T)| ≥2 in Propositions 3.5 and 3.6 of [7]. Therefore, if in the statement of Proposition 3.6 of [7]
we replace the(I−Sp)- and(T−(T∩Sp))-Grassmann geometries with the correspond- ing augmented Grassmann geometries, then we can drop the requirement|KSp(T)| ≥2 and obtain the following.
Proposition 3.2. SupposeBis a building of typeM overI. LetSp ⊆ I, letΓ be the augmented(I−Sp)-Grassmann geometry ofB, and letΣ = (P,L)be the point-line truncation ofΓ.
LetT ⊆I. LetCbe a chamber subsystem ofB, and supposeCis a chamber system of typeM|T. Let Cdenote the set of chambers ofC. ThenΣ|PC ∼= Σ(C), where the isomorphism takes a point or lineOofΣ|PCto the point or line ofΣ(C)corresponding to the residueRO∩CofC.
4 Proof of Theorem 1.1
First we prove Lemma 4.1, which is the main part of the proof of Theorem 1.1. For some motivations for the proof see Section 5.
We need the notion of retraction onto an apartment in a building. SupposeB is a building of typeM overI. LetAbe an apartment ofBand letcbe a chamber ofA. The retractionofBontoAwith centercis a mapretrA,c:B →Adefined as follows. Letx be a chamber ofB, and letwbe a geodesic fromctoxinB. Letw0be the unique gallery inAwhose initial chamber iscand whose type ist(w). ThenretrA,c(x)is defined to be the terminal chamber ofw0. By property (P) of buildingsretrA,c(x)does not depend on the choice of the geodesicwfromctox. The mapretrA,c : B → Ais a morphism of chamber systems overI, and its restriction toAis the identity map.
Suppose now thatBis a spherical building. LetAbe an apartment ofBand letcbe a chamber ofA. Supposexis a chamber ofB, and let gallerieswandw0 be as in the preceding paragraph. Letx0be the unique chamber ofAopposite toretrA,c(x). Thenc lies on a geodesicw00◦w0inAfromx0toretrA,c(x). The galleryw00◦w0is of reduced type, therefore the galleryw00◦wis of reduced type, and by property (G) of buildings the galleryw00◦wis a geodesic fromx0toxinB. In particular,x0is opposite toxinB.
Details can be found in Section 4.2 of [11].
Lemma 4.1. LetBbe a building of rank at least2, and suppose that the hypothesis of Theorem1.1holds. If only condition(C1)holds andrank(B)≥3, then assume that the implication of Theorem1.1is true for all residues ofBdifferent fromB.
LetAbe an apartment ofB. ThenhhPAii=P.
Proof. LetX= chamB(hhPAii). The lemma will be proved if we can show thatXis the set of all chambers ofBor, equivalently, if we can prove the following statement.
(S)For every chamberxofB, the chamberxand all the panels ofBonxare inX. First, we prove statement (S0).
(S0)Ify∈A, then all panels ofBonyare contained inX.
Let the setAebe defined as in Section 3. ThenAe ⊆ X. By Lemma 3.1 the setAe contains all panels ofBon every chamber ofA, therefore so doesX. This proves (S0).
Letc ∈Aand letxbe a chamber ofB. To prove statement (S) we use induction on d(c, x). Ifd(c, x) =0, thenx=cand by (S0) all panels ofBonxlie inX.
Supposed(c, x)≥1 and assume that, for every chamberyofBat distanced(c, x)−1 fromc, the chambery and all the panels ofB ony lie inX. Letabe a chamber ofB adjacent toxand such thatd(c, a) = d(c, x)−1. By the induction hypothesisa∈ X, and all panels ofBonaare contained inX. Thereforex∈X.
It remains to show that all panels ofBonxare contained inX. Letα∈Iand letQα be the panel of type{α}onx. We need to show thatQα⊆X.
Letp∈ Pbe such thatx∈Rp. We consider two cases separately.
Case1.α∈Sp. In this caseQα⊆Rp. Since the setXis a union of residues ofBof typeSp,x∈X, andRpis the unique residue of typeSpcontainingx, we haveRp⊆X.
ThereforeQα⊆X.
Case2. α∈ I−Sp. Letx0be the unique chamber ofAopposite toretrA,c(x)in A. Thenx0 is opposite toxinB. LetQbe the unique residue ofBcontainingx0 and opposite toRpinB. Letx00= gateQ(x), and letq∈ Pbe such thatx00∈Rq.
First, we are going to show thatq∈ hhPAii(step (2.1)). Then we use Proposition 2.6 to obtain a geodesicW fromxtox00inBwith certain properties (step (2.2)). Then we use the geodesicW to show thatQα⊆X.
(2.1)PQ⊆ hhPAii. In particular,q∈ hhPAii.
Suppose first that (C2) holds. Then the setsSp andI−Sp are stabilized byopM, thereforetyp(Q) =SpandQ= Rq. We remark that this includes, in particular, the case Sp=∅, that is the case when the points ofΓare the chambers ofB, and the lines ofΓare the panels ofB. SinceQ∩A6=∅, we haveq∈ PA.
Suppose now that (C1) holds. That is,|I−Sp|=1. Ifrank(B) =2, then|Sp|=1, andRpandQare panels ofB. Sincex0∈Q∩A, by (S0)Q⊆X. ThereforePQ⊆ PX. Supposerank(B)≥3. LetT = typ(Q). By (B3)Qis a building of typeM|T. Let
∆ = Σ|PQ = (PQ,L|PQ)be the geometry of points and lines ofΣlying entirely in PQ, and let∆0 = Σ(Q) = (P(Q),L(Q))be the point-line truncation of the augmented (T −(T ∩Sp))-Grassmann geometry ofQ. By Proposition 3.2 ∆ ∼= ∆0, where the isomorphism, which we denoteϕ, takes a point or lineOofΣ|PQto the point or line of Σ(Q)corresponding to the residueRO∩QofQ.
We havetyp(Q) = T = opM(Sp). SinceSp 6=I, the residueQis not the entire buildingB. Since the set Sp satisfies condition (C1), that is |I−Sp| = 1, we have
|T−(T∩Sp)| ≤1. Therefore, the setT∩Speither satisfies condition (C1) with respect to the diagramM|T, or T ∩Sp = T, that is T ∩Sp satisfies (C2) with respect to the diagramM|T. The intersection Q∩Ais an apartment ofQ(see Lemma 3.4 of [7] or
Lemma 2.4 of [6]), therefore by hypothesishh(P(Q))Q∩Aii∆0 = P(Q). Applying the isomorphismϕ−1toΣ(Q)we obtain thathhPQ∩Aii∆=PQ.
By Corollary 6.2 of [6] (or by Proposition 2.6 of [7]) the setPQis a convex subspace ofΣ. Therefore, the convex subspace closure ofPQ∩A in∆coincides with the convex subspace closure ofPQ∩AinΣ. That is,PQ =hhPQ∩Aii ⊆ hhPAii. This completes the proof of (2.1).
(2.2)There is a geodesicWfromxtox00inB, such that the first edge ofW is labelled {α}, and all vertices ofwG(W)are inhhPAii.
By the current hypothesisI−Sp 6=∅. That isSp 6= I, thereforeRp∩Q =∅ and p6=q. By Proposition 2.6 there is a geodesicz= (q0, q1, . . . , qn)fromp=q0toq=qn inG, such thatRpandRq1are connected by an edge labelled{α}.
LetZ be a gallery inBbeginning atxand ending atx00, such thatwG(Z) = z. By Corollary 2.2 there is a geodesicW fromxtox00such that eithert(W) = t(Z)ort(W) is obtained fromt(Z)by omitting one or more terms.
Letw = wG(W)and suppose thatw = (p0, . . . , pm), wherep0 = pandpm =q.
The walkzis a geodesic fromptoqinG, andl(w) ≤l(z). Thereforem= n, and no letters belonging toI−Spwere omitted when switching fromZtoW. In particular, the residuesRpandRp1are connected by an edge labelled{α}.
Let x1 be the chamber that follows xin the geodesic W. Since by its definition x00= gateRq(x), and the residuesQandRpare opposite inB, we havex= gateRp(x00).
The gallery W is a geodesic from x tox00 inB, thereforex1 6∈ Rp. It follows that x1∈Rp1, and{x, x1}is an edge labelled{α}.
Sincex∈X ∩Rp, we havep∈ hhPAii. Also, by (2.1)q ∈ hhPAii. The sethhPAii is convex inG, therefore all vertices of the geodesicwlie inhhPAii. This completes the proof of (2.2).
Let the geodesicsW andwbe as in 2.2. LetL = hp, p1i. By Lemma 3.10 of [7]
x, x1 ∈RLandα∈typ(RL). ThereforeQα∩RL 6=∅andtyp(Qα)⊆typ(RL). By property (B4) this implies thatQα⊆RL.
The sethhPAiiis a subspace ofΣ, and the lineLhas two pointspandp1 inhhPAii.
ThereforePL ⊆ hhPAii, wherePL denotes the set of all points ofΣincident withL.
SinceX = chamB(hhPAii), this implies chamB(PL) ⊆ X. ThereforeRL ⊆ X and Qα⊆X. This completes the proof of Case 2 and the proof of the lemma. 2 Proof of Theorem 1.1. Assume that the hypothesis of Theorem 1.1 holds. In particular, Bis a spherical building,Ais an apartment ofB, and the setSpsatisfies at least one of the conditions (C1) and (C2). We need to show thathhPAii=P.
Observe that ifrank(B) =0 (that is, ifBis a single chamber andΣis a single point), or ifrank(B) = 1 (that is,Bis a single panel andΣis either a single point or a single line), thenhhPAii=P. Therefore, in the rest of the proof we assume thatrank(B)≥2.
We consider two cases.
(1) |I−Sp|=1, that is condition (C1) holds. To show thathhPAii=Pwe use induction on the rank ofB, starting withrank(B) =2. The proofs of both, the initial induction step and the general induction step, are immediate from Lemma 4.1.
(2) The setI−Spis stabilized byopM, that is condition (C2) holds. ThenhhPAii=P
by Lemma 4.1. 2
5 Some remarks on the caseSp=∅
As has been mentioned before, ifBis a building (or any camber system) over a type setI, andSp=∅, then the points of the(I−Sp)-Grassmann geometry ofBare the chambers of Band the lines are the panels ofB. In this section we describe two results, Propositions 5.2 and Proposition 5.4, that relate to this case and that were the initial motivation for our proof of Theorem 1.1. For some other results pertaining to convex sets of chambers in a building see [1].
The proof of Lemma 4.1, Case 2, is reminiscent of the proof of Theorem 4.1.1 of [11]. Proposition 5.2 below can easily be proved using the same induction approach, and using Lemma 5.1. Combining Proposition 5.2 and Lemma 3.1 one immediately obtains Corollary 5.3, which is a special case of Theorem 1.1 corresponding toSp=∅, a special case of condition (C2).
Making an additional assumption that B is thick and repeating almost exactly the proof of Theorem 4.1.1 of [11] one obtains Proposition 5.4.
Lemma 5.1. LetBbe a spherical building of typeM overI, and letX be a convex set of chambers ofB. LetQandQ0 be opposite panels ofB. Suppose that Q ⊆ X and Q0∩X 6=∅. ThenQ0⊆X.
Proposition 5.2. LetBbe a spherical building of typeM overIand letAbe an apart- ment ofB. Suppose thatXis a convex set of chambers ofBcontainingAand such that, for every chamberx∈A, all panels ofBonxare contained inX. ThenX =B.
Corollary 5.3. LetBbe a spherical building of typeMoverI, and letAbe an apartment ofB. SupposeSp=∅and letΣ = (P,L)be the point-line truncation of the augmented (I−Sp)-Grassmann geometry ofB. ThenhhPAii=P.
Proof. Since Sp = ∅, the points of Σ are the chambers ofB, the lines ofΣ are the panels ofB, and the point-collinearity graph ofΣis the graph of the chamber systemB.
ThereforePA =AandPfA =A. This impliese Ae⊆ hPAi ⊆ hhPAii. By Lemma 3.1 the setAecontainsAand all the panels ofBon every chamber ofA, therefore so does the set hhPAii. Since the sethhPAiiis convex inB, by Proposition 5.2hhPAii=B. 2 Proposition 5.4. LetBbe a thick spherical building of typeM overI. SupposeAis an apartment ofB, and letcbe a chamber ofA. Suppose thatXis a convex set of chambers ofB, containingAand all the panels ofBonc. ThenX =B.
Proof. Letxbe a chamber ofB. We use induction ond(x, c)to show thatxand all the panels ofBonxare contained inB.
Ifx=c, then the statement is the hypothesis of the proposition. Supposed(x, c)≥1.
Letabe a chamber ofBadjacent toxand such thatd(c, a) = d(c, x)−1. Thenxis contained in a panel ofBona, therefore by the induction hypothesisx∈X. It remains to show that all the panels ofBonxare contained inX.
LetQ1 be the panel ofB onaandx, and letQ2 be any panel of Bonx. We can assume thatQ2 6=Q1, since otherwiseQ2 ⊆X by the induction hypothesis. Leta0 and
x0 be the unique chambers ofAopposite toretrA,c(a)andretrA,c(x)respectively. Let Q01be the panel ofBona0andx0. ThenQ1andQ01are opposite inB.
SinceBis thick, there is a chamberx00∈Q01opposite to bothaandx. Sincea0 ∈A, and by the induction hypothesisQ1 ⊆X, by Lemma 5.1 applied toQ1andQ01we have Q01 ⊆X. Thereforex00∈X.
LetQ02 be the panel ofBonx00opposite toQ2, and letQ002 be the panel ofBona opposite toQ02. By the induction hypothesisQ002 ⊆X, therefore by Lemma 5.1 applied to Q002 andQ02we obtain thatQ02⊆X. The chamberxis inX, and the panelQ2is opposite toQ02inB, therefore applying Lemma 5.1 toQ02andQ2we obtain thatQ2⊆X.
We have shown that, for every chamberxofB, the chamberxand all the panels ofB
onxare contained inX. ThereforeX =B. 2
6 Applications
In this section we describe two applications of Theorem 1.1. In Section 6.1 we charac- terize, under certain conditions, those convex subspaces of the augmentedJ-Grassmann geometry of a buildingBthat are isomorphic to shadows of spherical residues ofB(The- orem 6.2 and Corollaries 6.3 and 6.4). In Section 6.2 we show that, ifΣis the point- line truncation of the augmentedJ-Grassmann geometryΓof a building, then the point shadowX of a plane ofΓ cannot be properly contained in a subspace which is a gen- eralized polygon, provided that the point-collinearity graph ofΣ|X has finite diameter (Proposition 6.6).
6.1 A characterization of convex subspaces isomorphic to the shadow of a spherical residue. Suppose B is a building of type M over a type set I. Let J ⊆ I and let Σ = (P,L)be the point-line truncation of the augmentedJ-Grassmann geometry of B. Theorem 1.1 can be used together with Theorem 3.3 of [6] to characterize convex subspaces ofΣisomorphic to the subspaceΣ|PR, whereRis a spherical residue ofB. We formulate this result below as Theorem 6.2 (see also Corollaries 6.3 and 6.4). Theorem 6.2 will be proved using the approach used to prove Theorems 2.15, 3.6, and 4.7 of [3]. A different characterization of point shadows of residues ofB, similar to Theorem 3.3 of [6], will be given in a forthcoming paper.
To state and prove Theorem 6.2 we need the following notion. Suppose(P,L)is a point-line space with the set of pointsP and the set of linesL. Suppose that all singular subspaces of(P,L)are projective spaces. Let (P,E) be the point collinearity graph of(P,L). We say that a subsetX ofP issingularly independentors-independent in (P,L), if every finite subsetC of X such that the graph(P,E)|C is a clique spans a singular subspace of(P,L)of projective dimension|C| −1. We say that an induced subgraphGof(P,E)is s-independent, if its vertex set is s-independent.
Suppose now thatΣ = (P,L)is the point-line truncation of an augmentedJ-Grass- mann geometry of a buildingB. By Corollary 3.14 of [7] every singular subspace ofΣis a projective space, therefore the notion of s-independence can be used inΣ. The following lemma shows that the point shadow of an apartment ofBis always s-independent inΣ.
Lemma 6.1. LetBbe a building of typeM overI. LetSp ⊆Iand letΣ = (P,L)be the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofB. Suppose Ais an apartment ofB. Then the setPAis s-independent inΣ.
Proof. If|KSp(I)| =0, thenΣis a single point and the statement is obvious. Suppose that|KSp(I)| ≥1. LetGbe the point-collinearity graph ofΣ, and letCbe a clique ofG contained inPA. Suppose that|C|is finite. We need to show that the span ofCinΣis a singular space of projective rank|C| −1.
By Proposition 3.2 the mapϕ: Σ|PA →Σ(A), that takes a point or lineOofΣ|PA
to the point or line ofΣ(A)corresponding to the residueRO∩AofA, is an isomorphism of geometries. LetC0be the image ofCunderϕ.
Every line of the geometryΣ(A)is incident with exactly two points, therefore the cliqueC0is a subspace ofΣ(A). SinceC0is a singular subspace ofΣ(A)of finite projec- tive rank|C| −1, by Corollary 3.15 of [7] there is a residueA0ofAof typeT, for some T ⊆I, such thatC0 =P(A)A0. Moreover, the diagramM|KSp(T)is of typeA|C0|−1, and(I−Sp)∩KSp(T)is one of its end nodes.
LetQbe the residue ofBof typeT containingA0. LetQ0 be a residue ofQof type KSp(T). Using (B4) and (B5) we see thatΣ(Q)∼= Σ(Q0). Therefore, the geometryΣ(Q) is the point-line geometry of a building of typeA|C0|−1, corresponding to an end node of the diagram. By 6.3 of [11]Σ(Q)is a projective space of projective rank|C0| −1.
By Proposition 3.2 the mapψ: Σ|PQ →Σ(Q), that takes a point or lineOofΣ|PQ
to the point or line ofΣ(Q)corresponding to the residueQ∩ROofQ, is an isomorphism of geometries. By Proposition 2.6 of [7]PQ is a subspace of Σ, therefore Σ|PQ is a singular subspace ofΣof projective rank|C0| −1.
By Lemma 2.4 of [6] (or by Lemma 3.4 of [7])A0 is an apartment ofQ. Therefore, the setP(Q)A0 spans the projective spaceΣ(Q). SincePA0 = ψ−1(P(Q)A0), the set PA0 spansΣ|PQ. The cliqueCis contained inPAandϕ(C) =C0 =P(A)A0, therefore
C=PA0andCspansΣ|PQ. 2
Theorem 6.2. Suppose that Bis a building of type M over a type setI. LetSp ⊆ I and letΣ = (P,L)be the point-line truncation of the augmented(I−Sp)-Grassmann geometry ofB. LetG= (P,E)be the point-collinearity graph ofΣ.
SupposeT is a subset ofI such that the Coxeter diagramM|KSp(T)is spherical.
LetRbe a residue ofB of typeT, and letAbe an apartment of R. Suppose that the following condition holds.
(Apt)T For every s-independent induced subgraphG= (V, E)ofGisomorphic to the induced subgraphG|PA, there is an apartmentA0 of a residue ofBof typeT such thatV =PA0.
Then, for every convex subspace Σ1 = (P1,L1)ofΣ isomorphic to the subspace Σ|PR, there is a residueQofBof typeT such thatP1=PQandΣ1= Σ|PQ.
Proof. LetT,R, andAbe as in the hypothesis. LetΣ1= (P1,L1)be a convex subspace of Σand suppose ϕ : PR → P1 is an isomorphism of the subspace Σ|PR onto the subspaceΣ1. LetG1= (P1,E1)denote the point-collinearity graph ofΣ1.
Let V = ϕ(PA), and let G = (V, E) be the subgraph ofG1 induced onV. By Lemma 6.1P(R)A is s-independent inΣ(R), and by Proposition 3.2Σ|PR ∼= Σ(R).
