Expansion of Building-Like Complexes

Transcription

1 arxiv: v [math.co] 23 Jul 204 Expansion of Building-Like Complexes Alexander Lubotzky Roy Meshulam Shahar Mozes July 24, 204 Abstract Following Gromov, the coboundary expansion of building-like complexes is studied. In particular, it is shown that for any n, there exists a constant ǫ(n) > 0 such that for any 0 k < n the k-th coboundary expansion constant of any n-dimensional spherical building is at least ǫ(n). Introduction Expander graphs have been a focus of intensive research in the last four decades, with many applications in combinatorics and computer science as well as pure mathematics (see [6, 0, ]). In recent years a high dimensional theory is emerging. There are several ways to extend the definition of expanders from graphs to simplicial complexes (see [2] for a survey). Here we will be concerned with the notion of coboundary expansion that came up independently in the work of Linial, Meshulam and Wallach [9, 5] on homological connectivity of random complexes and in Gromov s work [5] on the topological overlap property. For an application of coboundary expansion to property testing see [7]. The rich theory of expander graphs hints that high dimensional expanders can also be useful. The goal of this paper is Institute of Mathematics, Hebrew University, Jerusalem 9904, Israel. alexlub@math.huji.ac.il. Supported by ERC, ISF and NSF grants. Department of Mathematics, Technion, Haifa 32000, Israel. meshulam@math.technion.ac.il. Supported by an ISF grant. Institute of Mathematics, Hebrew University, Jerusalem 9904, Israel. mozes@math.huji.ac.il. Supported by ISF and BSF grants

2 to show, following Gromov [5], that spherical buildings, and more generally, building-like complexes (defined precisely below), are expanders. We proceed with the formal definitions. Let X be a finite n-dimensional pure simplicial complex. For k 0, let X (k) denote the k-dimensional skeleton of X and let X(k) be the family of k-dimensional faces of X, f k (X) = X(k). Define a positive weight function w = w X on the simplices of X as follows. For σ X(k), let c(σ) = {η X(n) : σ η} and let w(σ) = ( n+ k+ c(σ) ) fn (X). Note that σ X(k) w(σ) = and if σ X(k) then w(τ) = (k +2)w(σ). {τ X(k+):σ τ} All homology and cohomology groups referred to in the sequel are with F 2 coefficients. Let C k (X) be the space of F 2 -valued k-chains of X with the boundary map k : C k (X) C k (X). Let C k (X) denote the space of F 2 - valued k-cochains of X with the coboundary map d k : C k C k+. As usual, the spaces of k-cycles and k-cocycles are denoted by Z k (X) and Z k (X) and the spaces of k-boundaries and k-coboundaries are denoted by B k (X) and B k (X). Reduced k-dimensional homology and cohomology will be denoted by H k (X) and H k (X). For φ C k (X), let [φ] denote the image of φ in C k (X)/B k (X). Let φ = w(σ) and {σ X(k):φ(σ) 0} [φ] = min{ φ+d k ψ : ψ C k (X)}. Definition.. The k-th coboundary expansion constant of X is { } dk φ h k (X) = min [φ] : φ Ck (X) B k (X). Remarks:. Note that h k (X) = 0 iff H k (X;F 2 ) Let n denote the n-simplex and let 0 k n. In [5, 5] it was shown that the k-th coboundary expansion of n satisfies h k ( n ) n+ n k 2 ()

3 with equality when n+ is divisible by k Let k < n and let σ X(k) be a k-simplex of minimal weight. Then [ σ ] = w(σ) and therefore h k (X) d k σ [ σ ] = {w(τ) : τ X(k +),σ τ} w(σ) = k +2. (2) Equality in (2) is attained for X = n and k = n. 4. The normalization we use for the norm in C k (X) and hence for the definition of h k (X) takes into account the possibility that the k-faces of X may not all have the same degrees. This is particularly relevant for spherical buildings - see the example following Corollary 3.6. In this note we are concerned with the expansion of certain building-like complexes. Let G be a subgroup of Aut(X) and let S be a finite G-set. For 0 k n, let F k = S X(k) with a G-action given by g(s,τ) = (gs,gτ). Let B = {B s,τ : k < n,(s,τ) F k } be a family of subcomplexes of X such that τ B s,τ B s,τ for all s S and τ τ X (n ). Definition.2. A building-like complex is a 4-tuple (X,S,G,B) as above with the following properties: (C) G is transitive on X(n). (C2) gb s,τ = B gs,gτ for all g G and (s,τ) S X (n ). (C3) H i (B s,τ ) = 0 for all (s,τ) F k and i k < n. Examples of building-like complexes include basis-transitive matroid complexes and spherical buildings - see Section 3. Following Gromov [5], we give a lower bound on the expansion of building-like complexes. For a simplex η X, let Gη denote the orbit of η under G. For 0 k n, let a k = a k (X,S,G,B) = max{ Gη B s,τ (k +) : η X(k +),(s,τ) F k }. Theorem.3. Let (X, S, G, B) be an n-dimensional building-like complex. Then for 0 k n, (( ) ) n+ h k (X) a k. (3) k +2 3

4 The proof of Theorem.3 is given in Section 2. In Section 3 we use Theorem.3 to derive expansion bounds for basis-transitive matroid complexes and for spherical buildings. In Section 4 we discuss applications to topological overlapping and to property testing. We conclude in Section 5 with some questions and comments. 2 A Lower Bound on Expansion Let (X,S,G,B) be a building-like complex. For a k-simplex τ = (v 0,...,v k ) and 0 i k, let τ i = (v 0,...,v i,v i+,...,v k ). The proof of Theorem.3 depends on the following homological filling property. Proposition 2.. There exists a family of chains such that C = {c s,τ C k+ (B s,τ ) : k n, (s,τ) F k } k+ c s,τ = τ + k c s,τi. (4) Proof: We define the c s,τ s by induction on k. First let k = and let be the empty simplex. For each s S, choose an arbitrary vertex v s B s, and let c s, = v s. For the induction step, let 0 k n and suppose that the c s,τ s have been defined for all (s,τ) j<k F j and that the family {c s,τ : (s,τ) F j, j < k} satisfies (4). Let (s,τ) F k. Then z = τ + k c s,τi C k (B s,τ )+ k C k (B s,τi ) C k (B s,τ ). We claim that z Z k (B s,τ ). Indeed k k z = k τ + k c s,τi k = τ i + k = c s,τij = 0. i,j (τ i + j c s,τij ) 4

5 The last equality follows from the fact that each c s,τij appears twice. As H k (B s,τ ) = 0, itfollowsthatthereexistsa(k+)-chainc s,τ C k+ (B s,τ )such that k+ c s,τ = z. It is clear that the family {c s,τ : (s,τ) F j, j k} satisfies (4). For 0 k n and s S, define the contraction operator ι s : C k (X) C k (X) as follows. For α C k (X) and τ X(k ) let ι s α(τ) = α(c s,τ ). Claim 2.2. For 0 k n and α C k (X) Proof: Let τ X(k). Then d k ι s α+ι s d k α = α. (5) d k ι s α(τ)+ι s d k α(τ) = ι s α( k τ)+d k α(c s,τ ) k = ι s α(τ i )+α( k+ c s,τ ) = = k α(c s,τi )+α(τ + k α(c s,τi )+α(τ)+ k c s,τi ) k α(c s,τi ) = α(τ). Remark: If α is a k-cocycle, then (5) gives a way of representing α as a k-cobounday, i.e. α = d k ι s α. For a general α C k (X), it provides a another representative of [α] C k (X)/B k (X). Proof of Theorem.3. Let 0 k n and α C k (X). Fix s S then by Claim 2.2, ι s d k α = α d k ι s α. Therefore [α] ι s d k α. (6) 5

6 For η X(k +), let and Let λ(η) = λ(η) = S w(η) S w(η) Proposition 2.3. For 0 k n, {(s,τ) F k :η supp(c s,τ)} {(s,τ) F k :η B s,τ} w(τ) w(τ). θ k = θ k (X,C) = max λ(η). (7) η X(k+) h k (X) θ k. (8) Proof: Let α C k (X). Summing (6) over all s S we obtain S [α] s S ι s d k α = s S {w(τ) : τ X(k), ιs d k α(τ) 0} = s S {w(τ) : τ X(k), dk α(c s,τ ) 0} s S {w(τ) : τ X(k), supp(dk α) supp(c s,τ ) } = = η supp(d k α) s S {w(τ) : τ X(k),η supp(cs,τ )} η supp(d k α) {(s,τ) F k :η supp(c s,τ)} η supp(d k α) S θ k S w(η)λ(η) η supp(d k α) = S θ k d k α. w(η) 6 w(τ)

7 To complete the proof of Theorem.3, it thus suffices to show the following: Claim 2.4. θ k ( ) n+ a k. (9) k +2 Proof: Fix an η X(k + ). By the homogeneity condition (C2), λ(η) = λ(gη) for all g G. The transitivity assumption (C) implies that f n (X) (G : stab G (η))c(η) and hence Therefore stab G (η) G c(η) f n (X) = G ( ) n+ w(η). k +2 G λ(η) G λ(η) = g G λ(gη) = g G = S w(gη) S w(η) S w(η) = a k stab G (η) w(η) {(s,τ) F k :gη B s,τ} w(τ) w(τ) {g G : gη B s,τ } (s,τ) F k (s,τ) F k w(τ) stab G (η) a k S = a k stab G (η) w(η) ( ) n+ G a k. k +2 (s,τ) F k w(τ) 3 Building-Like Complexes In this section we give applications of Theorem.3 to two families of buildinglike complexes. 7

8 3. Basis-Transitive Matroidal Complexes Let M be a matroid on the vertex set V with rank function ρ and let n = ρ(v). We identify M with its n-dimensional matroidal complex, namely the simplicial complex on V whose simplices are the independent sets of the matroid. Matroidal complexes are characterized by the property that their induced subcomplexes M[S] are pure for every S V. It is well known (see e.g. Theorem 7.8. in [2]) that H i (M) = 0 for all i < dimm = n. A matroid M is basis-transitive if its automorphism group Aut(M) is transitive on the bases (i.e. maximal faces) of M. One such example is the independence matroid of a vector space. For a classification of basistransitive matroids see [3] and the references therein. Let M be a basistransitive matroid of rank n + and let G be a subgroup of Aut(M) such that G is transitive on the facets. Let S = M(n) be the G-set of all n-faces of M. For (s,τ) S M(k) = F k let B s,τ = M[s τ]. Then gb s,τ = B gσ,gτ for all g G. As ρ(s τ) = n+, it follows that H i (B s,τ ) = H i (M[s τ]) = 0 for all i < n. Letting B = {B s,τ : k < n,(s,τ) F k } it follows that (M, S, G, B) is an n-dimensional building-like complex. Now ( ) n+k +2 a k = a k (M,S,G,B) max{f k+ (B s,τ ) : (s,τ) F k }. k +2 Writing ǫ (n,k) = (( n+ k+2 )( n+k+2 k+2 )), Theorem.3 implies the following Corollary 3.. If M is basis-transitive matroid of rank n + then for all 0 k n, h k (M) ǫ (n,k). Remark: The bound given in Corollary 3. is in general weak and can sometimes be significantly improved for specific classes of basis-transitive matroids by explicitly constructing a family of chains {c s,τ } satisfying (4) and then using Proposition 2.3 directly. We illustrate this by the following example. The Partition Matroid Let V,...,V n+ ben+disjoint sets such that V i = mandlet X = X n,m be the partition matroid with respect to V,...,V n+, i.e. σ X n,m iff σ V i for all i n+. Fix a vector v = (v,...,v n+ ) V = V V n+. 8

9 For an integer l let [l] = {,...,l}. Let k n and let τ = {u i : i I} X n,m (k) where u i V i and I ( [n+] k+). Define and let j = j(τ) = max{l : [l] I} τ = {u i : i [j]} X n,m (j ), τ = {u i : i I [j]} X n,m (k j). For T [j], let σ T = {v t : t T} {u t : t [j] T} and let z τ = T [j] σ T. If v i u i for all i [j], then z τ is the fundamental cycle of the octahedral (j )-sphere {u,v } {u j,v j }. Otherwise z τ = 0. Define c τ C k+ (X n,m ) as the concatination z τ v j+ τ. For i I, let τ i = τ {u i }. Claim 3.2. Proof: Note that for any i [j], As j z τ = 0 it follows that k+ c τ = τ + i I {T [j]:maxt=i} c τi. σ T τ = c τi. k+ c τ = k+ (z τ v j+ τ ) = z τ k+ j (v j+ τ ) = z τ τ + z τ v j+ (τ {u i }) i I [j] = z τ τ + i I [j] = τ τ + = τ + i [j] = τ + i [j] = τ + i I T [j] c τi σ T τ + {T [j]:maxt=i} c τi + c τi. i I [j] c τi i I [j] c τi σ T τ + i I [j] c τi 9

10 Keeping the notation j = j(τ), we next note that Therefore τ X n,m(k) { 2 j if u supp( c τ ) = t v t for all t [j] 0 otherwise. k+ supp( c τ ) = 2 j j=0 j=0 ( ( ) n j (m ) )m j k+ j k + j k+ ( ) j ( ) 2(m ) n j = m k+. m n k (0) () Let S = Aut(X n,m ) be the automorphism group of X n,m. For s S and τ X n,m (k), let c s,τ = s c sτ. Claim 3.2 implies that the family C = {c s,τ C k+ (B s,τ ) : k n, (s,τ) S X n,m (k)} satisfies (4). We proceed to compute θ k = θ k (X n,m,c) as defined in (7). First note that c s,s0 τ = s 0 c ss0,τ for all s 0,s S and τ X n,m (k). This, together with the transitivity of S on X n,m (k) and (), imply that for all s S τ X n,m(k) supp(c s,τ ) = τ X n,m(k) j=0 supp(c identity,τ ) = τ X n,m(k) k+ ( ) j ( ) 2(m ) n j = m k+. m n k supp( c τ ) (2) The transitivity of S on X n,m (k + ) implies that λ(η) is independent of η X n,m (k +). As λ(η) = S w(η) {(s,τ) F k :η supp(c s,τ)} w(τ) = f k+(x n,m ) S f k (X n,m ) {(s,τ) F k : η supp(c s,τ )}, 0

11 it follows by () that θ k = f k+ (X n,m ) S f k (X n,m ) ( n+ k+ j=0 η X n,m(k+) λ(η) = supp(c s,τ ) s S τ F k = k+ ( ) j ( 2(m ) n j ) m n k ). (3) Proposition 2.3 and (3) imply the following: Theorem 3.3. For 0 k n, h k (X n,m ) k+ ( n+ ) k+ j=0 (2(m ) m We note some special cases of Theorem 3.3. (i) Let m =. Then X n, = n and ( n+ ) k+ h k (X n, ) ( n ) = n+ n k k+ thereby recovering the bound (). (ii) Let m = 2. Then X n,2 is the octahedral n-sphere and ) ) h k (X n,2 ) k+ j=0 ( n+ k+ ( n j n k ) = )j( ). (4) n j n k ( n+ k+ ( n+ n k ) =. This coincides with the result of Proposition 5.5 in [4]. (iii) For general m and k = n, h n (X n,m ) n+ n j=0 (2(m ))j. m This is a small improvement over the bound h n (X n,m ) n+ Proposition 5.7 in [4]. 2 n+ given in We conclude this section with an upper bound on the expansion of X n,m.

12 Claim 3.4. Let 0 k n. If (k +2) m then h k (X n,m ). Proof: Let V i = k+2 j= V ij where V ij = m. Let α k+2 Ck (X n,m ) be the indicator function of the following set of k-simplices: V i,π() V ik+,π(k+). i < <i k+ n+π S k+ Then supp(α) = The support of the coboundary of α is B = supp(d k α) = and so, B = ( )( ) k+ n+ m (k +)!. k + k +2 i < <i k+2 n+ π S k+2 V i,π() V ik+2,π(k+2) ( )( ) k+2 n+ m (k +2)!. k +2 k +2 We claim that [α] = α. Indeed, suppose that α = α + d k ψ where ψ C k (X n,m ). Let C = {(σ,τ) supp(α ) B : σ τ}. As B = supp(d k α) = supp(d k α ), it follows that C B. On the other m hand, any τ X n,m (k) is contained in at most (n k) simplices of B. k+2 It follows that B C supp(α m ) (n k) k +2. Therefore It follows that supp(α k +2 ) B m(n k) = supp(α). d k α [α] = f k(x n,m ) supp(d k α) f k+ (X n,m ) supp(α) ( n+ ) )( k+ m k+ m k+2 (n+ k+2 = k+2) (k +2)! ) )( m k+2 m ) k+(k =. (n+ +)! ( n+ k+2 k+ 2 k+2

13 3.2 Spherical Buildings In this section we use Theorem 3 to recover Gromov s [5] uniform lower bound on the expansion of spherical buildings of rank n+. Our notation and terminology follows []. Let G = B,N be a finite group with a BN-pair of rank n+ and let (W,S) be the associated Coxeter system. Here W = N/(B N)istheWeyl groupandsisthedistinguishedset ofn+generators of W. For J S, let W J = J and let G J = BW J B be the associated standard parabolic group. For s S, let (s) = S {s}. The spherical building = (G;B,N) is the n-dimensional pure simplicial complex on the vertex set V = s S G/G (s) whose maximal faces, called chambers, are C g = {gg (s) : s S}. Two chambers are adjacent if their intersection is (n )-dimensional. For g G, let V g = {gwg (s) : w W,s S}. The apartment A g is the induced complex [V g ]. It is a simplicial n-sphere whose chambers are {C gw } w W, hence f n (A g ) = W. Any two simplices σ,τ are contained in some apartment A g. Claim 3.5. Let g,...,g k G and let Y = k i= A g i. If dimy = n then H i (Y) = 0 for all i n. Proof: It is convenient to identify the complex with its geometric realization. Recall the following: A gallery connecting two simplices σ and τ is a sequence C 0,C,...,C r of adjacent chambers so that σ is a face of C 0 and τ is a face of C r. The gallery is called minimal if it has minimal length among all possible galleries connecting σ and τ. An apartment A which contains two simplices contains also every minimal gallery connecting them. Let x and y be two points in the geometric realization of. Let A be an apartment containing x and y. Consider the sequence of consecutive chambers visited by a minimal geodesic on the sphere A connecting x and y. This sequence forms a minimal gallery and hence is contained in any apartment containing both x and y. Fix some x Y = k i= A g i. If Y contains a point antipodal to x in some apartment A 0 containing xthen itfollows fromtheabove that Y contains the apartment A 0 and hence it follows that Y = A 0 which is an n-dimensional 3

14 sphere and the claim holds. Otherwise it follows that for each y Y there is a unique geodesic arc connecting x to y in all the apartments containing x and y and in particular this geodesic arc is contained in Y. This implies that Y is contractible. Let S = (n) be the set of chambers of. For (s,τ) S (k) = F k, let B s,τ = {A g : s,τ A g }. Letting B = {B s,τ : k < n,(s,τ) F k } it follows from Claim 3.5 that (,S,G,B) satisfies conditions (C),(C2) and (C3) of Definition.2. Clearly ( ) ( ) n+ n+ a k = a k (,S,G,B) f k+ (A g ) f n (A g ) = W. k +2 k +2 Let ω n be the maximal size of a Weyl group of rank n+ and let Theorem.3 then implies ( (n+ ) ) 2 ǫ 2 (n,k) = ω n. k +2 Corollary 3.6. If G = B,N is a finite group with BN-pair of rank n+, then for all 0 k n h k ( (G;B,N)) ǫ 2 (n,k). Example: Let G = GL n+2 (F q ) = B,N where B is the group of upper diagonal matrices and N is the group of monomial matrices. The Weyl group of G is the symmetric group W = S n+2. The n-dimensional spherical building = (G;B,N), denoted by A n+ (F q ), is isomorphic to the order complex of all nontrivial linear subspaces of F n+2 q. Corollary 3.6 implies that for 0 k n, ( (n+ ) ) 2 h k (A n+ (F q )) (n+2)!. (5) k +2 4

15 In particular h n (A n+ (F q )) (n+2)!. (6) Remark: The uniform lower bound (5) on the expansion of A n+ (F q ) depends on the particular normalization used in the definition of the norm in C k (X). Indeed, (5) fails to hold if the weight of a k-simplex is simply taken f k (X) as. For example, let k = 0 and fix an n such that n+2 be divisible by 2. If U is an n+2 -dimensional subspace of Fn+2 2 q, then the degree of U in the underlying graph G of A n+ (F q ) is at most This is much smaller than 2f 0 (An (F q)) = q (n+2) 2 2 f (A n+ (F q )) f 0 (A n+ (F q )) = q (n+2) 2 3 (+o()) q (n+2)2 (+o()) 4 6 (+o()). = q (n+2)2 2 (+o()). It follows that the -dimensional skeleton of A n+ (F q ) is not an expander if one uses the normalization giving the same weight to all i-simplices. 4 Applications Lower bounds on coboundary expansion give rise to applications in two directions: topological overlapping and property testing. 4. Topological Overlapping Let X be a finite n-dimensional pure simplicial complex and let M be an n-dimensional Z 2 -manifold. For a continuous map f : X M and a point p M, let γ f (p) = {σ X(n) : p f(σ)}. The following result is due to Gromov [5]. See also [4] for a detailed exposition(includingsomeimprovedconstants) forthecasem = R n andx = (n) N. Theorem 4. ([5]). For any ǫ > 0 there exists a δ = δ(m,ǫ) > 0 such that if h k (X) ǫ for all 0 k n, then there exists a point p M such that γ f (p) δf n (X). 5

16 Let (X,S,G,B) be an n-dimensional building-like complex and let a(x,s,g,b) = max 0 k n a k(x,s,g,b). The following consequence of Theorem.3 was already noted by Gromov (section 2.3 in [5]) when X is a spherical building or a partition matroid. Corollary 4.2. For any 0 < c and an n-dimensional Z 2 -manifold M, there exists a constant δ = δ(c,m) > 0 such that if a(x,s,g,b) c, then for any continuous map f : X M there exists a point p M such that γ f (p) δf n (X). 4.2 Property Testing Definition 4.3. Let A be a finite set, and let dist(, ) be a metric on A m. Let W m a subset of A m and P m a subset of W m. Let ǫ > 0 and q N be fixed. We say that the membership of α P m (given α W m ) is (q,ǫ)-testable, if there exists a randomized algorithm which queries only q (independent of m) coordinates of α and answers yes if α P m, while it answers no with probability at least ǫ dist(α,p m ). In [7], it was observed that coboundary expansion implies that the subspace of coboundaries is testable within the subspace of cochains. The distance function dealt with there was the Hamming distance, but the same applies to the norm used the this paper, provided the algorithm chooses a face with probability equal to its norm. Theorem.3 therefore implies the following. Corollary 4.4. Forany0 < c andk < n there existanǫ = ǫ(c,k,n) > 0 such that if an n-dimensional building-like complex satisfies a k (X,S,G,B) c, then checking whether a k-cochain α is a k-coboundary is (k +2,ǫ)-testable. 5 Concluding Remarks We mention some problems related to the results of this paper.. It would be interesting to improve the bounds given in Theorem.3 and its corollaries. One concrete question is the following. The - dimensional building A 2 (F q ) is the points vs. lines graph of the Desarguian projective plane of order q. It is known that the normalized 6

17 Cheegerconstantofthisgraphsatisfiesh 0 (A 2 (F q )) = o()asq. It seems likely that for n 2 the bound(6) can similarly be improved. Conjecture 5.. For fixed n and q h n (A n+ (F q )) = o(). 2. Let L n be a geometric lattice of rank n with minimal element 0 and maximal element. Let X(L n ) be the order complex of L n { 0, }. Then X(L n ) is (n 2)-dimensional and H k (X(L n )) = 0 for k n 3 (see e.g. [2]). It would be interesting to find natural families {L n } for which h n 3 (X(L n )) remains uniformly bounded away from zero. For example, is this the case when L n is the lattice of partitions of [n+3]? ACKNOWLEDGEMENT The authors would like to thank Jake Solomon for helpful discussions. References [] A. Björner, Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings Adv. in Math. 52(984) [2] A. Björner, Topological methods, in Handbook of Combinatorics (R. Graham, M. Grötschel, and L. Lovász, Eds.), , North- Holland, Amsterdam, 995. [3] A. Delandtsheer and H. Li, Basis-transitive matroids, J. Algebraic Combin. 3(994) [4] D. Dotterrer and M. Kahle, Coboundary expanders, J. Topol. Anal. 4(202) [5] M. Gromov, Singularities, expanders and topology of maps. Part 2: From combinatorics to topology via algebraic isoperimetry, Geom. Funct. Anal. 20(200) [6] S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications, Bull. Amer. Math. Soc. (N.S.) 43(2006)

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