A MOORE BOUND FOR SIMPLICIAL COMPLEXES

Transcription

1 Bull. London Math. Soc. 39 (2007) C 2007 London Mathematical Society doi: /blms/bdm003 A MOORE BOUND FOR SIMPLICIAL COMPLEXES ALEXANDER LUBOTZKY and ROY MESHULAM Abstract Let X be a d-dimensional simplicial complex with N faces of dimension d 1. Suppose that every (d 1)-face of X is contained in at least k d + 2 faces of X, of dimension d. Extending the classical Moore bound for graphs, it is shown that X must contain a ball B of radius at most log k d N whose d-dimensional homology H d (B) is non-zero. The Ramanujan complexes constructed by Lubotzky, Samuels and Vishne are used to show that this upper bound on the radius of B cannot be improved by more than a multiplicative constant factor. 1. Introduction Let G =(V,E) be a graph on n vertices, and let δ(g) denote the minimal degree of a vertex in G. The girth g(g) =g is the minimal length of a cycle in G. An easy counting argument (see, for example, [2, Theorem IV.1]) shows that if δ(g) =k 3, then 1+ k ( (k 1) (g 1)/2 1 ) g odd, k 2 n 2 ( (k 1) g/2 1 ) (1.1) g even. k 2 This implies the classical Moore bound. Theorem A. g(g) < 2 log k 1 n +2. Let d G (u, v) be the distance between the vertices u and v in the graph metric, and let B r (v) ={u V : d G (u, v) r} denote the ball of radius r around v. Define the acyclicity radius r v (G) ofg at the vertex v to be the maximal r such that the induced graph G[B r (v)] is acyclic. Let r(g) = min v V r v (G); then r(g) = g(g)/2 1. The asymptotic version of Moore s bound is equivalent to the following statement. Theorem A 1. If δ(g) =k 3, then for every v V r v (G) log k 1 n. (1.2) The best lower bound for the girth of k-regular graphs is given by the Ramanujan graphs of Lubotzky, Phillips and Sarnak [6]. For a fixed prime p, the construction given in [6] provides a sequence of (p + 1)-regular graphs G i =(V i,e i ) with V i such that g(g i ) 4 3 log p V i O(1). A similar result was obtained by Morgenstern [9] for any prime power q. In terms of the acyclicity radius we therefore have our next theorem. Received 28 October 2005; revised 4 July 2006; published online 16 March Mathematics Subject Classification 20E42 (primary), 13F55 (secondary). The first author is supported by the Ambrose Monell Foundation, the Ellentuck Fund and the Israel Science Foundation. The second author is supported by a State of New Jersey grant, and by the Israel Science Foundation.

2 354 ALEXANDER LUBOTZKY AND ROY MESHULAM Theorem B 1. For every prime power q, there exists a sequence of (q +1)-regular graphs G i =(V i,e i ) with V i such that for every v V, r v (G i ) 2 3 log q V i O(1). In this paper we extend Theorems A 1 and B 1 to higher-dimensional simplicial complexes. Let X be a d-dimensional simplicial complex on the vertex set V. Let H i (X) denote the i- dimensional homology group of X with some fixed field coefficients. For 0 i d let X(i) = {σ X : dim σ = i}, and let f i (X) = X(i). For a subset of vertices S V, let X[S] denote the induced subcomplex on S. The degree ofa(d 1)-simplex σ X(d 1) is deg(σ) = {τ X(d) :σ τ}. Let δ(x) = min{deg(σ) :σ X(d 1)}. A complex X is called k-regular if deg(σ) =δ(x) =k for every σ X(d 1). Denote by B r (v) the ball of radius r around v with respect to the graph metric on the one-dimensional skeleton of X. Extending the notion of acyclicity radius to the higher-dimensional setting, we define r v (X) as the maximal r such that H d (X[B r (v)]) = 0, and r(x) = min v V r v (X). The following result extends Theorem A 1 to d-dimensional complexes. Theorem A d. Let X be a d-dimensional complex with δ(x) =k d +2. Then for any vertex v V which is contained in some (d 1)-face, r v (X) log k d f d 1 (X). For the lower bound, we use the Ramanujan complexes presented by Lubotzky, Samuels and Vishne in [8] to show the following statement. Theorem B d. For d 1 and q a prime power, there exists a sequence of d-dimensional (q +1)-regular complexes X i on vertex sets V i with V i, such that for any v V r v (X i ) log q V i 2d 2 (d +2) 1. Theorem A d is proved in Section 2, while Theorem B d is established in Section 3. Note that Theorem A d reduces to Theorem A 1 when d = 1. On the other hand, specializing theorem B d for the case d = 1 yields a somewhat weaker version of Theorem B 1 (the constant is 1 6 rather than 2 3 ). In Section 4 we discuss some open problems and suggestions for further research. One such challenge is to improve the constant in Theorem B d. 2. The upper bound Proof of Theorem A d. First note that if Y is a d-dimensional complex such that f d (Y ) > f d 1 (Y ), then H d (Y ) 0. Indeed, let C i (Y ) denote the space of simplicial i-chains of Y. Then dim C d (Y )=f d (Y ) >f d 1 (Y ) = dim C d 1 (Y ) implies that the boundary map : C d (Y ) C d 1 (Y ) has a non-trivial kernel. Let v be a vertex which is contained in a (d 1)-simplex. Abbreviate B t = B t (v) and write α(t) =f d 1 (X[B t ]), β(t) =f d (X[B t ]). Let Then γ(t) = {(σ, τ):σ X[B t ](d 1), τ X(d), σ τ}. γ(t) = σ X[B t](d 1) deg(σ) f d 1 (X[B t ]) δ(x) α(t) k. (2.1)

3 A MOORE BOUND FOR SIMPLICIAL COMPLEXES 355 For a d-simplex τ X(d), let s(τ) denote the number of (d 1)-simplices in X[B t ] that are contained in τ. Then { d +1 τ X[B t ], s(τ) = 0 τ X[B t+1 ], and s(τ) 1ifτ X[B t+1 ] X[B t ]. Thus γ(t) = s(τ) τ X(d) (d +1)β(t)+(β(t +1) β(t)) = dβ(t)+β(t +1). (2.2) Let m = r v (X). Combining (2.1) and (2.2), we find that, for all t<m, Hence kα(t) dβ(t)+β(t +1) dα(t)+α(t +1). α(t +1) (k d)α(t)... (k d) t α(1). Since v is contained in a (d 1)-face, it follows that α(1) kd +1.Thus and m log k d f d 1 (X). (kd + 1)(k d) m 1 α(m) f d 1 (X) 3. The lower bound The proof of Theorem B d depends on certain finite quotients of affine buildings constructed by Lubotzky, Samuels and Vishne [8], based on the Cartwright Steger group [4] (see also [11] for a similar construction, as well as [3, 5, 7] for related results). In Section 3.1 we recall the definition and some properties of affine buildings of type Ãd 1. In Section 3.2 we describe the relevant finite quotients, and show that they have a large acyclicity radius Affine buildings of Type Ãd 1 Let F be a local field with a valuation ν : F Z and a uniformizer π. Let O denote the ring of integers of F, and let O/πO = F q be the residue field. A lattice L in the vector space V = F d is a finitely generated O-submodule of V such that L contains a basis of V. Two lattices L 1 and L 2 are equivalent if L 1 = λl 2 for some 0 λ F. Let [L] denote the equivalence class of a lattice L. Two distinct equivalence classes [L 1 ] and [L 2 ] are adjacent if there exist representatives L 1 [L 1 ], L 2 [L 2 ] such that πl 1 L 2 L 1. The affine building of type Ãd 1 associated with F is the simplicial complex B = B d (F ) whose vertex set B 0 is the set of equivalence classes of lattices in V, and whose simplices are the subsets {[L 0 ],...,[L k ]} such that all pairs [L i ], [L j ] are adjacent. It can be shown that {[L 0 ],...,[L k ]} forms a simplex if and only if there exist representatives L i [L i] such that πl k L 0... L k. (3.1) It is well known that B is a contractible (d 1)-dimensional simplicial complex, and that the link of each vertex is isomorphic to the order complex A d 1 (F q ) of all non-trivial proper linear subspaces of F d q (see, for example, [10, 7]). This implies that δ(b) =q + 1. The type function τ : B 0 Z d is defined as follows. Let O d be the standard lattice in V. For any lattice L, there exists g GL(V ) such that L = go d. Define τ([l]) = ν(det(g))(mod d). Let dist([l], [L ]) denote the graph distance between [L], [L ] B 0 in the 1-skeleton of B. Let dist 1 ([L], [L ])

4 356 ALEXANDER LUBOTZKY AND ROY MESHULAM denote the minimal t for which there exist [L] =[L 0 ],...,[L t ]=[L ] such that [L i ] and [L i+1 ] are adjacent in B and τ([l i+1 ]) τ([l i ]) = 1 for all 0 i t 1. Claim 3.1. For two lattices L 1 and L 2, dist 1 ([L 1 ], [L 2 ]) (d 1) dist([l 1 ], [L 2 ]). (3.2) Proof. This follows directly from inclusion (3.1). Alternatively, let v 1,...,v d be a basis of V, and let a 1,...,a d be integers such that L 1 = d i=1 Ov i and L 2 = d i=1 πai Ov i. Then dist([l 1 ], [L 2 ]) = max a i min a i (3.3) i i and d dist 1 ([L 1 ], [L 2 ]) = a i d min a i. (3.4) i Now (3.2) follows from (3.3) and (3.4) Finite quotients of affine buildings Let q be a prime power, and let F be the local field F q ((y)) with local ring O = F q [[y]]. The construction of finite quotients of B = B d (F )in[8] depends on the remarkable Cartwright Steger group Γ < PGL d (F ) (see [4]). We briefly recall the construction of Γ and some of its properties. Let φ : F q d F q d denote the Frobenius automorphism. Extend φ to F q d(y) by defining φ(y) =y. Then φ is a generator of the cyclic Galois group Gal(F q d(y)/f q (y)). Let D be the d 2 - dimensional F q (y)-algebra given by D = F q d[σ] with the relations σa = φ(a)σ for all a F q d(y), and σ d =1+y. D is a division algebra that splits over the extension field F = F q ((y)). Denote D(F )= D F ; then there is an isomorphism D(F ) = M d (F ) which in turn induces an isomorphism D(F ) /Z(D(F ) ) = PGL d (F ). (3.5) Let b 1 =1 σ 1 D, and for u F q let b d u = u 1 b 1 u. Let g u D(F ) /Z(D(F ) ) denote the image of b u under the quotient map. The Cartwright Steger group Γ is the subgroup of D(F ) /Z(D(F ) ) generated by {g u : u F q }. Utilizing the isomorphism (3.5), we also regard d Γ as a subgroup of PGL d (F ). We shall use the following properties of Γ. i=1 Theorem 3.2 (Cartwright and Steger [4]). (a) Γ acts simply transitively on the vertices of B. (b) Let L 0 = O d. Then, for g Γ, dist 1 (g[l 0 ], [L 0 ]) = min{t : g = g u1 g ut for some u 1,...,u t F q d}. The action of D upon itself by conjugation gives rise to a representation ρ : D(F ) GL d 2(F ) which factors through D(F ) /Z(D(F ) ). Let ξ 0,...,ξ d 1 be a normal basis of F q d over F q ; then {ξ i σ j } d 1 i,j=0 is a basis of D(F ) over F. An explicit computation [8, p. 975, equation (9)] shows that k 1 b u (ξσ k )b 1 u = ξσ k u + f u,k (ξ) φ i (u) σi + 1 y f d 1 u u,k(ξ) φ i (u) σi i=0 i=0 for every 0 k<dand ξ F q d, where f u,k (ξ) =(φ k u/u)ξ (φ k 1 u/φ 1 u)φ 1 ξ.

5 A MOORE BOUND FOR SIMPLICIAL COMPLEXES 357 It follows that, with respect to the above basis, ρ(b u )isad 2 d 2 matrix whose entries are linear polynomials in 1/y over F q. Let h(λ) F q [λ] be an irreducible polynomial which is prime to λ(1 + λ), and let f = h(1/y) R 0 = F q [1/y] and I = fr 0. Write 1 d 2 for the d 2 d 2 identity matrix. Let Γ(I) ={γ Γ:ρ(γ) 1 d 2(mod f)}. This subgroup coincides with the congruence subgroup Γ(I) as defined in [8, p. 979, equation (15)]. In particular, Γ/Γ(I) is isomorphic to a subgroup of PGL d (R 0 /fr 0 ). Let B I =Γ(I) \B denote the resulting quotient complex. The vertex set BI 0 of B I is the set of orbits of B 0 under Γ(I); that is, BI 0 = {Γ(I)[L] :[L] B 0 }. A subset {Γ(I)[L 0 ],...,Γ(I)[L k ]} forms a simplex in B I if and only if there exist g 0,...,g k Γ(I) such that {g 0 [L 0 ],...,g k [L k ]} is a simplex in B. Note that BI 0 =(Γ:Γ(I)) PGL d (R 0 /fr 0 ). Let L be a lattice, and let l I = min{dist([l],g[l]):1 g Γ(I)}. Clearly, l I is independent of L since Γ is transitive and Γ(I) Γ. Proposition 3.3. We have l I log q B 0 I (d 1)(d 2 1). (3.6) Proof. Let t = dist 1 (g[l 0 ], [L 0 ]). By Theorem 3.2(b) there exist u 1,...,u t F q such that d g = g u1 g ut. Let C =(c ij )=ρ(b u1 ) ρ(b ut ). The c ij are polynomials in F q [1/y] of degree at most t in 1/y. By assumption, g Γ(I), and hence C = 1 d 2 + fe for some E M d 2(R 0 ). If c ij 0 for some i j, then t deg 1/y (c ij ) deg 1/y (f). Otherwise, C is a diagonal matrix. If it is a scalar matrix then it must be the identity, as Γ, being a lattice in PGL d (F ), has trivial center. Thus we can assume that C is diagonal and non-scalar. Choose i, j such that c ii c jj ; then t deg 1/y (c ii c jj ) deg 1/y (f). Thus, by (3.2) dist([l 0 ],g[l 0 ]) 1 d 1 dist 1(g[L], [L]) deg 1/y(f) (d 1) log q PGL d (R 0 /fr 0 ) (d 1)(d 2 1) log q BI 0 (d 1)(d 2 1). Proof of Theorem B d 1. Choose a sequence of irreducible polynomials h i (λ) F q [λ] such that (h i,λ(1 + λ)) = 1 and deg h i. Let I i = h i (1/y)R 0, and let X i = B Ii. The quotient map B X i is clearly an isomorphism on balls of radius at most (l Ii /2) 1inB. Since B is contractible, it follows from Proposition 3.3 that for any vertex v Xi 0, r v (X i ) l I i 2 1 log q X 0 i 2(d 1)(d 2 1) 1.

6 358 A MOORE BOUND FOR SIMPLICIAL COMPLEXES We complete the proof by noting that if i is sufficiently large, then l Ii 4, and hence X i is (d 1)-dimensional and δ(x i )=δ(b) =q Concluding remarks We have proved a higher-dimensional extension of the Moore bound, and have shown that the Ramanujan complexes constructed in [8] imply that this bound is tight up to a multiplicative factor. We mention some problems that arise from these results. (1) In Section 3.2 it is shown that for appropriately chosen ideals I i F q [1/y], the (d 1)- dimensional quotient complexes X i = B Ii satisfy r v (X i ) C(d 1) log q X 0 i 1 with C(d 1)=1/2(d 1)(d 2 1). It seems likely that a more careful choice of the I i will lead to an improved bound on the constant. (Recall that in the one-dimensional case, Ramanujan graphs [6] give the constant 2 3, while C(1) = 1 6.) (2) While the construction of Ramanujan graphs and the proof of Theorem B 1 depend on number-theoretic tools, there is an elementary (but non-constructive) argument due to Erdős and Sachs (see, for example, [1, Theorem III.1.4]) that shows the existence of a sequence of k-regular graphs G i =(V i,e i ) with V i such that r(g i ) 1 2 log k 1 V i O(1). It would be interesting to obtain a similar result in the higher-dimensional setting. References 1. B. Bollobás, Extremal graph theory (Academic Press, London, 1978). 2. B. Bollobás, Modern graph theory (Springer, New York, 1998). 3. D. Cartwright, P. Solé and A. Zuk, Ramanujan geometries of type A n, Discrete Math. 269 (2003) D. Cartwright and T. Steger, A family of Ãn-groups, Israel J. Math. 103 (1998) W.-C. W. Li, Ramanujan hypergraphs, Geom. Funct. Anal. 14 (2004) A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988) A. Lubotzky, B. Samuels and U. Vishne, Ramanujan complexes of type A d, Israel J. Math 49 (2005) A. Lubotzky, B. Samuels and U. Vishne, Explicit constructions of Ramanujan complexes of type A d, European J. Combin. 26 (2005) M. Morgenstern, Existence and explicit constructions of q + 1 regular Ramanujan graphs for every prime power q, J. Combin. Theory Ser. B 62 (1994) M. Ronan, Lectures on buildings, Perspectives in Mathematics 7 (Academic Press, London 1989). 11. A. Sarveniazi, Ramanujan (n 1,n 2,...,n d 1 )-regular hypergraphs based on Bruhat Tits buildings of type Ãd 1, arxiv.org/math.nt/ Alexander Lubotzky Institute for Advanced Study Princeton, NJ USA Current address: Institute of Mathematics Hebrew University Jerusalem Israel alexlub@math.huji.ac.il Roy Meshulam Institute for Advanced Study Princeton, NJ USA Current address: Department of Mathematics Technion Haifa Israel meshulam@math.technion.ac.il