Spectrum and combinatorics of two-dimensional Ramanujan complexes
SSPECTRUM AND COMBINATORICS OF TWO-DIMENSIONALRAMANUJAN COMPLEXES
KONSTANTIN GOLUBEV AND ORI PARZANCHEVSKI
Abstract.
Ramanujan graphs have extremal spectral properties, which imply a remark-able combinatorial behavior. In this paper we compute the high dimensional Hodge-Laplace spectrum of Ramanujan triangle complexes, and show that it implies a combina-torial expansion property, and a pseudo-randomness result. For this purpose we prove aCheeger-type inequality and a mixing lemma of independent interest. Introduction
Expanders are graphs whose nontrivial adjacency spectrum is concentrated in a narrow strip.This implies remarkable combinatorial properties, such as isoperimetric expansion [AM85],pseudo-randomness [AC88, FP87], rapid convergence of random walk and a large chromaticnumber [Hof70]. We refer to the surveys [HLW06, Lub12] for the applications of expandersin mathematics and computer science.A k -regular graph is called Ramanujan if its nontrivial spectrum is contained within the L -spectrum of its universal cover, which is the k -regular tree T k : Spec (cid:0)
Adj T k (cid:1) = (cid:104) − √ k − , √ k − (cid:105) (the trivial spectrum is, by definition, the eigenvalues ± k ). By the Alon-Boppana theorem(cf. [HLW06, Thm. 2.7]), this is asymptotically the best one can hope for, so that Ramanujangraphs are optimal expanders. Such graphs were first constructed in [LPS88, Mar88], asquotients of the Bruhat-Tits tree associated with P GL ( Q p ) by arithmetic lattices. It wassuggested by several authors [CSŻ03, Li04, LSV05a, Sar07] that Ramanujan complexes shouldbe defined as quotients of Bruhat-Tits buildings whose spectral properties agree with those ofthe building. The cited papers show that such complexes do exist, and the papers [FGL + P GL d , and these correspond torepresentations of the Hecke algebra of the group, which is commutative [Mac79]. In thispaper, we investigate the spectrum of the high-dimensional Hodge-Laplace operators, whichencode the homology of the complex in all dimensions [Eck44]. To achieve this, we interpret(see Proposition 3.1) the simplicial boundary and coboundary maps on Ramanujan complexesas intertwining maps between different representations of the (non-commutative) Iwahori-Hecke algebra of P GL d .For Ramanujan complexes of dimension two, we compute the Hodge-Laplace spectra in alldimensions, and show that unlike the situation in graphs, the nontrivial spectrum in dimension Date : June 6, 2018. a r X i v : . [ m a t h . C O ] J u l AMANUJAN TRIANGLE COMPLEXES 2 one is concentrated in two narrow strips. We give here a loose version, and a tight one appearsin Theorem 2.3.
Theorem 1.1.
Let X be a Ramanujan complex of type (cid:101) A , and ∆ i = δ ∗ δ + δδ ∗ the simplicialHodge-Laplace operator in dimension i ∈ { , , } (see definitions in §2). Then the nontrivialspectrum of ∆ i is contained in ∆ : S ∆ : S ∪ S , ∆ : { } ∪ S , where S = (cid:104) k − (cid:112) k − , k + 3 (cid:112) k − (cid:105) S = (cid:104) k − (cid:112) k − , k + 2 (cid:112) k − (cid:105) (cid:91) [2 k − , k + 8] , and k (resp. k ) is the vertex degree (resp. edge degree) in X . In the second part of the paper (Section 4) we explore the combinatorial information whichis encoded in the Hodge-Laplace spectrum. The results apply to any complex, and notonly to quotients of Bruhat-Tits buildings, so that Section 4 can be read independently. In§4.1 we prove the following theorem, which generalizes the isoperimetric inequalities from[AM85, PRT16]:
Theorem 1.2.
Let X be a d -dimensional complex on n vertices, and Z i = Z i ( X ) the space of i -dimensional cycles. If Spec ∆ i (cid:12)(cid:12) Z i ⊆ [ k i − µ i , k i + µ i ] for ≤ i ≤ d − and Spec ∆ d − (cid:12)(cid:12) Z d − ⊆ [ λ d − , ∞ ) , then for any partition Verts ( X ) = (cid:96) di =0 A i | X ( A , . . . , A d ) | n d | A | · . . . · | A d | ≥ k · . . . · k d − · λ d − (cid:32) − µ d − k d − − C d (cid:18) µ k + . . . + µ d − k d − (cid:19) n d +1 (cid:81) di =0 | A i | (cid:33) , where X ( A , . . . , A d ) are the d -cells of X in A × . . . × A d , and C d depends only on d . For a complex with a complete skeleton, one has k i = n and µ i = 0 for ≤ i ≤ d − , hencethe r.h.s. above reads as n d − λ d − , recovering Theorem 1.2 in [PRT16].In Theorem 4.1 we prove a generalization of the Expander Mixing Lemma (cf. [HLW06,§2.4]), showing that concentration of the Hodge-Laplace spectrum implies a pseudo-randombehavior. Combining these combinatorial theorems with Theorem 2.3, we obtain the follow-ing results on Ramanujan complexes of type (cid:101) A , which show that they enjoy isoperimetricexpansion, pseudo-randomness and large chromatic number. Theorem 1.3.
Let X be a Ramanujan complex of type (cid:101) A with n vertices, vertex degree k = 2 (cid:0) q + q + 1 (cid:1) and edge degree k = q + 1 . Fix a constant ϑ > .(1) (Isoperimetry) If X is not tripartite, then for any partition of the vertices of X intosets A , A , A of sizes at least ϑn, | X ( A , A , A ) | n | A | | A | | A | ≥ q − q . − C · q ϑ . (2) (Pseudo-randomness) If X is tripartite, let A, B, C, D be disjoint sets of vertices suchthat each of A ∪ D , B and C is contained in a different block of the tripartition of X . If A, B, C and D are of sizes at most ϑn , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( A, B, C, D ) (cid:12)(cid:12) − q | A | | B | | C | | D | n (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:0) q . ϑ + 244 q . (cid:1) ϑn AMANUJAN TRIANGLE COMPLEXES 3 where X ( A, B, C, D ) are the pairs of triangles t ∈ A × B × C , t ∈ B × C × D whichshare an edge ( | t ∩ t | = 2 ).(3) If X is not tripartite, the chromatic number of X is at least √ q . Here the chromatic number is the minimal number of colors needed to color the vertices of thecomplex with no monochromatic triangle. It is interesting to compare the last result with thatof [EGL15], which studies mixing in Ramanujan complexes using the spherical representationsalone (which correspond to operators on the vertices of the complex). They show that thechromatic number of such a complex is at least √ q , and we expect that in higher dimensionsour new methods should lead to an even greater advantage over the spherical analysis. Acknowledgement.
We thank Alex Lubotzky for providing us guidance and inspiration, andto Uriya First and Dima Trushin for patiently introducing us to the representation theory of p -adic groups. We are grateful to Shai Evra, Mark Goresky, Alex Kontorovich and Winnie Lifor helpful discussions and suggestions. The first author was supported by the ERC and bythe Israel Science Foundation. The second author was supported by The Fund for Math andby NSF grant DMS-1128155, and is grateful for the hospitality of the Institute for AdvancedStudy. 2. Complexes and buildings
We recall the basic elements of simplicial Hodge theory, and of Bruhat-Tits buildings of type (cid:101) A d . For a more relaxed exposition of the former we refer to [PRT16, §2], and for the latterto [Li04, LSV05a, Lub14].2.1. Simplicial Hodge Theory.
For a finite simplicial complex X of dimension d , denoteby X i the set of cells of dimension i in X ( i -cells ). The degree of an i -cell is the num-ber of ( i + 1) -cells which contain it. Denote by Ω i = Ω i ( X ) the space of i -forms, namelyskew-symmetric complex functions on the oriented i -cells, equipped with the inner prod-uct (cid:104) f, g (cid:105) = (cid:80) σ ∈ X i f ( σ ) g ( σ ) . The i -th boundary map ∂ i : Ω i → Ω i − is defined by ( ∂ i f ) ( σ ) = (cid:80) v : vσ ∈ X i f ( vσ ) , its dual is the i -th coboundary map δ i = ∂ ∗ i : Ω i − → Ω i ,given by ( δ i f ) ( σ ) = (cid:80) ij =0 ( − j f ( σ \ σ j ) , and Z i = ker ∂ i , Z i = ker δ i +1 , B i = im ∂ i +1 and B i = im δ i are the cycles, cocycles, boundaries and coboundaries, respectively.The upper, lower and full i -Laplacians are ∆ + i = ∂ i +1 δ i +1 , ∆ − i = δ i ∂ i and ∆ i = ∆ + i + ∆ − i ,respectively. Their spectra are closely related: Spec ∆ + i coincides with Spec ∆ − i +1 , up to adifference in the the multiplicity of zero (which is determined by the number of i -cells and i + 1 -cells), and the spectrum of ∆ i is, up to zeros, the union of Spec ∆ + i and Spec ∆ − i . It ismost convenient for our purposes to work with the upper Laplacian, and ∆ +0 is the classicalgraph Laplacian: (cid:0) ∆ +0 f (cid:1) ( v ) = deg ( v ) f ( v ) − (cid:88) w ∼ v f ( w ) . (2.1)2.2. Bruhat-Tits buildings.
Let F be a nonarchimedean local field with ring of integers O , uniformizer π , and residue field O / π O of size q , which we identify with F q . We denoteby B = B d ( F ) the building of type (cid:101) A d − associated with F , which is defined as follows.The vertices of B are the left K -cosets in G , where G = P GL d ( F ) and K = P GL d ( O ) .Each vertex gK is associated with the homothety class of the O -lattice g O d . A collection of AMANUJAN TRIANGLE COMPLEXES 4 vertices { g i K } i =0 ..r forms an r -cell if, possibly after reordering, there exist representatives g (cid:48) i ∈ GL d ( F ) for g i , such that πg (cid:48) O d < g (cid:48) r O d < g (cid:48) r − O d < . . . < g (cid:48) O d < g (cid:48) O d . (2.2)The group G acts on B by left translation, and if Γ is a torsion-free lattice in G then thequotient X = Γ \B is a finite complex. In the case d = 2 , the building B ( Q p ) is a ( p + 1) -regular tree, and its quotients by lattices in G = P GL ( Q p ) are ( p + 1) -regular graphs.Certain lattices give rise to Ramanujan quotients: Theorem 2.1 ([LPS88, Mar88], cf. [Sar90, Lub94]) . If Γ is a congruence subgroup of atorsion-free arithmetic lattice in G , then Γ \B is a Ramanujan graph. Namely, its spectrum is contained within {− p − } ∪ (cid:2) − √ p, √ p (cid:3) ∪ { p + 1 } , where the eigen-values ± ( p + 1) are the trivial ones: p +1 corresponds to the constant function on the vertices,and if the graph Γ \B is bipartite, − ( p + 1) appears as an eigenvalue of the function whichtakes one value on one side and the opposite value on the other.2.3. Trivial spectrum of complexes.
The function τ = ord π det : P GL d ( F ) → Z / d Z induces a d -partition on the vertices of B d . We define the type of a cell σ in B d to be thesubset { τ ( v ) | v ∈ σ } of Z / d Z , and say that a function on X i = Γ \B id is trivial if its lift to B id isconstant on each type. An eigenvalue of ∆ i (or of ∆ ± i ) is called trivial if it is obtained froma trivial eigenfunction, and thus the nontrivial spectrum of these operators is obtained fromtheir restriction to the functions which sum to zero on each type.In dimension two, the Bruhat-Tits building B has constant vertex and edge degrees k = 2 (cid:0) q + q + 1 (cid:1) and k = q + 1 respectively, and τ partites the vertices of B into three parts. We say that X = Γ \B istripartite when Γ preserves this partition. In this case the trivial eigenfunctions on verticesare Eigenfunction Eigenvalue of ∆ +0 Eigenvalue of ∆ − (cid:12)(cid:12) X (cid:12)(cid:12) v (cid:55)→ ω τ ( v ) , ω = e ± πi k When X is not tripartite, only the constant function is in the trivial spectrum. The edgesof B have a canonical orientation by setting τ (term ( e )) ≡ τ (orig ( e )) + 1 ( mod , and wecan thus define a form in f ∈ Ω ( B ) by assigning a value to the positively oriented edges.Furthermore, the action of Γ always preserves this orientation, so that the same holds for X = Γ \B . ( † ) The trivial eigenforms in Ω ( X ) for a tripartite X areEigenform on positive direction Eigenvalue of ∆ +1 Eigenvalue of ∆ − k e (cid:55)→ ω τ (orig( e )) , ω = e ± πi k and again, in the non-tripartite case only the constant one appears. ( † ) Namely, X is always disorientable , see [PR17, Def. 2.6]. AMANUJAN TRIANGLE COMPLEXES 5
Ramanujan complexes.
There are several plausible ways to define what are Ramanu-jan complexes, and these are discussed in [CSŻ03, Li04, LSV05a, KLW10, Fir16]. However,it can be shown that they all agree for complexes of type (cid:101) A , and amount to the following. Definition 2.2.
The complex X = Γ \B is Ramanujan if the nontrivial spectrum of theLaplace operators in every dimension is contained within that of the corresponding Laplaceoperators on L ( B ) .The papers [Li04, LSV05b, Sar07] give several constructions of Ramanujan complexes, someof which are the clique complexes of Cayley graphs.The following theorem determines the spectrum of the upper Laplacians on two-dimensionalRamanujan complexes. The spectra of the full and lower Laplacians can be inferred from it(see §2.1). By definition, this is the same as determining the L -spectrum of the Laplacians on B itself, but in addition we determine the multiplicity of eigenvalues on the finite quotients. Theorem 2.3.
Let X be a Ramanujan quotient of B with n vertices, and vertex and edgedegrees k = 2 (cid:0) q + q + 1 (cid:1) and k = q + 1 . If X is non-tripartite, then(1) ∆ +0 has the trivial eigenvalue , and n − nontrivial eigenvalues in [ k − q, k + 3 q ] .(2) ∆ +1 has(a) The trivial eigenvalue k .(b) n − zeros, corresponding to B ( X ) (coboundaries).(c) For every nontrivial λ ∈ Spec ∆ +0 , the eigenvalues k ± (cid:113)(cid:0) k (cid:1) − λ .This amounts to n − eigenvalues in each of the strips I − = (cid:20) k − (cid:113)(cid:0) k (cid:1) + 8 q, k + 1 (cid:21) I + = (cid:20) k − , k + (cid:113)(cid:0) k (cid:1) + 8 q (cid:21) . (2.3) (d) n (cid:0) q + q − (cid:1) + 2 eigenvalues in the strip I = [ k − √ q, k + 2 √ q ] . (2.4) If X is tripartite, then(1) ∆ +0 has trivial spectrum (cid:8) , k , k (cid:9) (see 2.3), and n − eigenvalues in [ k − q, k + 3 q ] .(2) ∆ +1 has(a) The trivial eigenvalue k , and two trivial zeros (both coming from B ( X ) ).(b) n − nontrivial zeros, all coming from B ( X ) .(c) n − eigenvalues in each of I ± , corresponding to k ± (cid:113)(cid:0) k (cid:1) − λ for λ a nontrivialeigenvalue of ∆ +0 .(d) n (cid:0) q + q − (cid:1) + 6 eigenvalues in I . Let us make a few remarks:(1) The spectrum of ∆ +0 is well-known [Mac79, Li04, LSV05a], but our methods are different,and give the spectrum in all dimensions in a unified manner.(2) These bounds are sharp: a sequence of Ramanujan complexes with injectivity radiusgrowing to infinity (as constructed in [LM07]) has Laplace spectra which accumulate toany point in these intervals. This follows from [Li04] for dimension zero and from [PR17,§3.5] for general dimension. AMANUJAN TRIANGLE COMPLEXES 6 (3) All the zeros in the spectra of ∆ +0 and ∆ +1 come from B ( X ) , so that the zeroth andfirst Betti numbers of X vanish, in accordance with [Gar73, Cas74].It is interesting to compare Theorem 2.3 with Garland’s spectral bounds: Theorem (Garland’s bound, [Gar73, Pap08, GW13]) . If X is a finite complex such that Spec ∆ +0 (link ( σ )) (cid:12)(cid:12) Z (link( σ )) ⊆ [ λ, Λ] for every σ ∈ X j − and k ≤ deg σ ≤ K for every σ ∈ X j − , then Spec ∆ + j ( X ) (cid:12)(cid:12) Z j ( X ) ⊆ [( j + 1) λ − jK, ( j + 1) Λ − jk ] . The links of vertices in B are incidence graphs of projective planes over F q , which have λ = k − √ q and Λ = 2 k , so that Garland’s bound implies that any quotient of B satisfies Spec ∆ +1 (cid:12)(cid:12) Z ⊆ [ k − √ q, k ] . Theorem 2.3 shows that both ends are tight! The drawback of Garland’s method is that itmisses the sparse picture within this interval, which is crucial for our combinatorial purposes,namely, the results in §4.1 and §4.2. The proof of Theorem 2.3 occupies the next threesections. 3.
Computation of the Laplace spectrum
Boundary maps as Iwahori-Hecke operators.
In this section we translate the sim-plicial boundary and coboundary maps into intertwining operators between different represen-tations arising from the group
P GL . Keeping the notations of §2.2, we fix the “fundamental”vertex v = K in B = G / K . It follows from the fact that Γ is torsion-free that it acts freelyon vertices, and thus if we normalize the Haar measure on G so that µ ( K ) = 1 , we have µ (Γ \ G ) = n . Furthermore, this implies a linear isometry Ω ( X ) ∼ = L (Γ \ G/K ) , given ex-plicitly by f ( gv ) = f (Γ gK ) . We identify L (Γ \ G/K ) with L (Γ \ G ) K , the space of K -fixedvectors in the G -representation L (Γ \ G ) .The element σ = (cid:16) π (cid:17) ∈ G acts on B by rotation on the triangle consisting of the vertices v , σv , and σ v . We fix the oriented edge e = [ v , σv ] , and define E = stab G e = K ∩ σKσ − = (cid:110)(cid:16) ∗ ∗ ∗∗ ∗ ∗ x y ∗ (cid:17) ∈ K (cid:12)(cid:12)(cid:12) x, y ∈ π O (cid:111) . (3.1)Since G acts transitively on the non-oriented edges of B , and preserves the canonical orienta-tion from §2.3, the positively oriented edges of X correspond to double cosets Γ \ G/E , givingan identification of Ω ( X ) with L (Γ \ G ) E , by f ( ge ) = f ([ gv , gσv ]) = (cid:112) µ ( E ) f (Γ g ) , f ([ gσv , gv ]) = − (cid:112) µ ( E ) f (Γ g ) , (3.2)where µ ( E ) = µ ( K )[ K : E ] = q + q +1 . The scaling by (cid:112) µ ( E ) is needed to make the isomorphism Ω ( X ) ∼ = L (Γ \ G ) E an isometry: if { g i e } nk / i =1 represent the edges of X and f ∈ Ω ( X ) then (cid:107) f (cid:107) ( X ) = (cid:88) i | f ( g i e ) | = (cid:88) i µ ( E ) | f (Γ g i ) | = (cid:90) Γ \ G | f (Γ g ) | dg = (cid:107) f (cid:107) L (Γ \ G ) . We fix the triangle t = (cid:2) v , σv , σ v (cid:3) , whose pointwise stabilizer is the Iwahori subgroup I = K ∩ σKσ − ∩ σ Kσ − = (cid:110)(cid:16) ∗ ∗ ∗ x ∗ ∗ y z ∗ (cid:17) ∈ K (cid:12)(cid:12)(cid:12) x, y, z ∈ π O (cid:111) . AMANUJAN TRIANGLE COMPLEXES 7
As for edges, G acts transitively on non-oriented triangles, and preserves triangle orientation.Thus, the stabilizer of t as a cell is T := stab G t = (cid:104) σ (cid:105) I = I (cid:116) σI (cid:116) σ I, and in particular (cid:104) σ (cid:105) and I commute. Again, f ( gt ) = (cid:112) µ ( T ) f (Γ g ) gives a linear isometry Ω ( X ) ∼ = L (Γ \ G ) T , where µ ( T ) = 3 µ ( I ) = µ ( K )[ K : I ] = q + q +1)( q +1) .Denoting K = K , K = E , and K = T , we have Ω i ( X ) ∼ = L (Γ \ G ) K i . ( † ) As I ≤ E ≤ K and I ≤ T , the three spaces L (Γ \ G ) K i are contained in L (Γ \ G ) I . The Iwahori-Heckealgebra H = C c ( I \ G/I ) consists of the compactly supported, bi- I -invariant complex functionson G , with multiplication defined by convolution. If ( ρ, V ) is a representation of G , then (cid:0) ρ, V I (cid:1) is a representation of H , where ρ ( η ) v := (cid:82) G η ( g ) ρ ( g ) v dg .We proceed to show that the (co-)boundary maps between L (Γ \ G ) K i are given by certainintertwining elements in H . Proposition 3.1.
The following elements of H : ∂ = √ µ ( E ) ( Kσ − K ) δ = √ µ ( E ) ( σK − K ) ∂ = √ µ ( E ) µ ( T ) · ET δ = √ µ ( E ) µ ( T ) · T E (3.3) act as the corresponding simplicial operators. Namely, each ∂ i ∈ H takes L (Γ \ G ) K i to L (Γ \ G ) K i − and acts as the boundary operator ∂ i : Ω i ( X ) → Ω i − ( X ) with respect to theidentifications of Ω i ( X ) with L (Γ \ G ) K i , and likewise for δ i ∈ H and δ i : Ω i − → Ω i .Proof. Both ∂ i +1 and δ i map any representation V into V K i , since they are constant on right K i cosets (note that σK = EσK ). Let f ∈ L (Γ \ G ) E ∼ = Ω ( E ) , and let K = (cid:96) kE ∈ K / E kE .For any gv ∈ X we have ( K f ) ( gv ) = ( K f ) (Γ g ) = (cid:90) G K ( x ) ( xf ) (Γ g ) dx = (cid:90) K ( xf ) (Γ g ) dx = (cid:90) K f (Γ gx ) dx = (cid:88) kE ∈ K / E (cid:90) E f (Γ gke ) de = (cid:88) kE ∈ K / E (cid:90) E f (Γ gk ) de = µ ( E ) (cid:88) kE ∈ K / E f (Γ gk ) = (cid:112) µ ( E ) (cid:88) kE ∈ K / E f ( gke ) . The group K acts transitively on the q + q + 1 positive edges leaving v , so that the positiveedge leaving gv (for any g ∈ G ) are { gke } kE ∈ K / E . Therefore, (cid:112) µ ( E ) ( K f ) ( gv ) = (cid:88) kE ∈ K / E f ( gke ) = (cid:88) orig e = gv e positive f ( e ) = − (cid:88) term e = gv e negative f ( e ) . (3.4) ( † ) This nice picture only holds for
P GL . In P GL d with d ≥ the group does not act transitively on eachdimension, and there are also elements which flip orientations in the middle dimension. AMANUJAN TRIANGLE COMPLEXES 8
In a similar manner, the positive edges with terminus gv are (cid:8) gkσ e (cid:9) k ∈ K , and if Kσ E = (cid:96) kσ E ∈ Kσ E / E kσ E then ( Kσ f ) ( gv ) = (cid:90) Kσ E f (Γ gx ) dx = (cid:88) kσ E ∈ Kσ E / E (cid:90) E f (cid:0) Γ gkσ e (cid:1) de = (cid:112) µ ( E ) (cid:88) kσ E ∈ Kσ E / E f (cid:0) gkσ e (cid:1) = (cid:112) µ ( E ) (cid:88) term e = gv e positive f ( e ) . (3.5)Together with (3.4), this implies that ∂ from (3.3) indeed act as the simplicial ∂ , justifyingthe abuse of notation. The reasoning for ∂ is similar, save for the fact that T (cid:2) E (infact, E ∩ T = I ). However, E acts transitively on the triangles containing e , hence for f ∈ L (Γ \ G ) T √ µ ( E ) µ ( T ) ( ET f ) ( ge ) = √ µ ( T ) ( ET f ) (Γ g ) = (cid:112) µ ( T ) (cid:88) eT ∈ ET / T f (Γ ge )= (cid:88) eT ∈ ET / T f ( get ) = (cid:88) τ ∈ X : e ∈ ∂τ f ( gτ ) = (cid:88) τ ∈ X : ge ∈ ∂τ f ( τ ) , agreeing with ∂ : Ω → Ω . The coboundary operators can be analyzed in a similar manner,or as follows: H is a ∗ -algebra by η ∗ ( g ) = η ( g − ) , and a unitary representation ρ of G induces a unitary H -representation, i.e. ρ ( η ) ∗ = ρ ( η ∗ ) (this uses unimodularity of G ). For V = L (Γ \ G ) this gives ∂ ∗ = √ µ ( E ) (cid:16) ( Kσ ) − − K − (cid:17) = √ µ ( E ) ( σK − K ) and similarly for ∂ ∗ . (cid:3) Since Γ is cocompact, L (Γ \ G ) decomposes as a sum of irreducible unitary representations, L (Γ \ G ) = (cid:76) α W α , and Ω i ( X ) ∼ = L (Γ \ G ) K i = (cid:76) α W K i α ≤ (cid:76) α W Iα . Each W Iα is a sub- H -representation, so that the operators ∂ i , δ i decompose with respect to this sum, and thusthe Laplacians as well, giving Spec∆ ± i = (cid:83) α Spec ∆ ± i (cid:12)(cid:12) W Kiα , with the correct multiplicities.To understand the spectra it is enough look at the W α which are Iwahori-spherical, namely,contain I -fixed vectors. Furthermore, the isomorphism type of W α already determines thespectrum of ∆ ± i on W K i α . By [Cas80, Prop. 2.6], if W Iα (cid:54) = 0 then W α is embeddable in a principal series representation V z . Namely, there exists z = ( z , z , z ) ∈ C (the Satakeparameters ) with z z z = 1 , and V z = uInd GB χ z = (cid:110) f : G → C (cid:12)(cid:12)(cid:12) f ( bg ) = δ − ( b ) χ z ( b ) f ( g ) ∀ b ∈ B (cid:111) , (3.6)where χ z is the character χ z ( b ) = (cid:81) i =1 z ord π b ii i of the Borel group B := (cid:110)(cid:16) ∗ ∗ ∗ ∗ ∗ ∗ (cid:17) ∈ G (cid:111) , and δ ( b ) = | b | / | b | is the modular character of B . For obvious reasons, it is convenient tointroduce the notation (cid:101) χ z ( b ) = δ − ( b ) χ z ( b ) = | b || b | (cid:89) i =1 z ord π b ii i = (cid:18) z q (cid:19) ord π b z ord π b ( qz ) ord π b . Having decomposed L (Γ \ G ) = (cid:76) α W α , and found a ∆ ± i -eigenform f ∈ W K i α ≤ Ω i ( X ) , wecan lift it to a Γ -periodic eigenform ˜ f ∈ Γ Ω i ( B ) . For some z we have Ψ : W α (cid:44) → V z , andnaturally Ψ f ∈ V K i z ; since V z is defined as a set of complex functions on G , we can think of AMANUJAN TRIANGLE COMPLEXES 9 Ψ f as an i -form on B . Thus, both ˜ f and Ψ f are ∆ ± i -eigenforms with the same eigenvalue.However, they are not the same, as ˜ f attains finitely many values and Ψ f infinitely many, ingeneral. Nevertheless, the matrix coefficients g (cid:55)→ (cid:68) g (cid:101) f , (cid:101) f (cid:69) and g (cid:55)→ (cid:104) g Ψ f, Ψ f (cid:105) are the same,since Ψ is a unitary embedding. When these matrix coefficient are in L ε ( G ) for every ε > , and only then, the representation W α is weakly contained in L ( G ) , which impliesthat the corresponding eigenvalue is in the L -spectrum of ∆ ± i on B (cf. [CHH88]).3.2. Analysis of the principal series.
Even though W α is only a subrepresentation of V z ,it is simpler to consider ∂ i , δ i and ∆ ± i acting on V z , and later restrict to W α . The Weyl groupof G is S (as permutation matrices), and G decomposes as G = BK = (cid:97) w ∈ A BwE = BT (cid:116) B (1 2) T = (cid:97) w ∈ S BwI.
From G = BK and (3.6) we see that dim V K z ≤ , and in fact this is an equality since χ z (cid:12)(cid:12) B ∩ K ≡ , hence f K ( bk ) := (cid:101) χ z ( b ) is well defined. Similarly, dim V I z = 6 , with basis (cid:8) f Iw (cid:9) w ∈ S defined by f Iw ( w (cid:48) ) = δ w,w (cid:48) , and dim V E z = 3 with basis (cid:8) f Ew (cid:9) w ∈ A , where f Ew := f Iw + f Iw · (1 2) satisfies f Ew ( w (cid:48) ) = δ w,w (cid:48) for w, w (cid:48) ∈ A . Finally, dim V T z = 2 with basis f Tw ( w (cid:48) ) := δ w,w (cid:48) for w, w (cid:48) ∈ { ( ) , (1 2) } , which satisfy f T ( ) = f I ( ) + 1 qz f I (3 2 1) + z q f I (1 2 3) f T (1 2) = f I (1 2) + z f I (2 3) + 1 qz f I (1 3) ; (3.7)indeed, if c w is the coefficient of f Iw in f T ( ) , then c (1 2 3) = f T ( ) ((1 2 3)) = f T ( ) (cid:16)(cid:16) π (cid:17) σ (cid:17) = f T ( ) (cid:16)(cid:16) π (cid:17)(cid:17) = (cid:101) χ z (cid:16)(cid:16) π (cid:17)(cid:17) = z q , and the other coefficients in (3.7) are obtained similarly.Let Ω i z ( B ) be the realization of V K i z as a subspace of Ω i ( B ) , given by the explicit construction(3.6). Any f ∈ Ω z ( B ) is determined by its value on v , namely f = f ( v ) f K . Similarly, thevalue on e , e := (3 2 1) e and e := (1 2 3) e determine a unique element in Ω z ( B ) , andlikewise for t , t := (1 2) t and Ω z ( B ) . As S ≤ K , one can compute the action of ∂ i (cid:12)(cid:12) Ω i z ( B ) and δ i (cid:12)(cid:12) Ω i − z ( B ) by evaluation on star ( v ) alone (see Figure 3.1). Figure 3.1.
The star of v in B . AMANUJAN TRIANGLE COMPLEXES 10
For the basis B E = (cid:110) f E ( ) , f E (3 2 1) , f E (1 2 3) (cid:111) one has (cid:104) δ (cid:12)(cid:12) Ω z ( B ) (cid:105) { f K } B E = qz − z − z q − ; (3.8)for example, (cid:0) δ f K (cid:1) ( e ) = f K ((3 2 1) σv ) − f K ( v ) = f K (cid:16)(cid:16) π (cid:17) v (cid:17) − z − , and (cid:0) δ f K (cid:1) ( e ) , (cid:0) δ f K (cid:1) ( e ) are computed similarly. We turn to ∂ . The positive edges withorigin v are e , (cid:16) x (cid:17) e for x ∈ F q = O / π O , and (cid:16) x y (cid:17) e with x, y ∈ F q . As (cid:101) χ z istrivial on upper-triangular unipotent matrices, (3.4) implies that for f ∈ Ω z ( B ) we have (cid:16) µ ( E ) − K f (cid:17) ( v ) = f ( e ) + qf ( e ) + q f ( e ) . The positive edges entering v are (cid:2) σ v , v (cid:3) = σ e = (cid:16) π π (cid:17) e = (cid:16) π π (cid:17) (1 2 3) e = (cid:16) π π (cid:17) e , and similarly (cid:16) π x π (cid:17) e and (cid:16) π xπ y (cid:17) e ( x, y ∈ F q ). By (3.5), (cid:18) √ µ ( E ) Kσ E f (cid:19) ( v ) = f (cid:16)(cid:16) π π (cid:17) e (cid:17) + (cid:88) x ∈ F q f (cid:16)(cid:16) π x π (cid:17) e (cid:17) + (cid:88) x,y ∈ F q f (cid:16)(cid:16) π xπ y (cid:17) e (cid:17) = z qz f ( e ) + q · z z f ( e ) + q · z q z f ( e ) , and in total (see (3.3)) (cid:104) ∂ (cid:12)(cid:12) Ω z ( B ) (cid:105) B E { f K } = (cid:0) qz − qz − q qz − q (cid:1) . (3.9)As ∆ +0 = ∂ δ and ∆ − = δ ∂ , we can now compute explicitly their action on the z -principalseries. Denoting (cid:101) z = (cid:80) i =1 (cid:0) z i + z − i (cid:1) , we have by (3.8) and (3.9) ∆ +0 (cid:12)(cid:12) Ω z ( B ) = (cid:0) λ K (cid:1) := ( k − q (cid:101) z ) , (3.10) (cid:104) ∆ − (cid:12)(cid:12) Ω z ( B ) (cid:105) B E = q − qz − qz +1 − q z + q z z + q − qz − q z + q z z + q − qz qz z − z − qz +1 − qz +2 q − qz − q z + q + qz z − qz − qz − z q + z z +1 q − z − qz + z z q − qz − qz +1 . Observe that ∆ +0 agrees with the computation of the spectrum of the Hecke operators in[Mac79, CS02, Li04, LSV05a], as ∆ +0 = k · I − (cid:80) d − i =1 A i , where A i is the i -th Hecke operatoron B d (loc. cit.). In effect, ∆ − can also be understood without the machinery above, as ithas the eigenvalue λ K corresponding to δ Ω z ( B ) (since ∆ − δ f K = δ ∆ +0 f K = λ K δ f K ),and two zeros (which come from ∂ Ω z ( B ) ). However, this machinery allows us to computeas easily the edge/triangle spectrum: for f ∈ Ω z ( B ) , one has ( δ f ) ( t ) = (cid:88) i =0 f (cid:0) σ i e (cid:1) = f ( e ) + f (cid:16)(cid:16) π (cid:17) (3 2 1) e (cid:17) + f (cid:16)(cid:16) π π (cid:17) (1 2 3) e (cid:17) = f ( e ) + qz f ( e ) + z qz f ( e )( δ f ) ( t ) = (cid:88) i =0 f (cid:0) (1 2) σ i e (cid:1) = f ( e ) + f (cid:16)(cid:16) π (cid:17) (1 2 3) e (cid:17) + f (cid:16)(cid:16) π π (cid:17) (3 2 1) e (cid:17) = f ( e ) + qz f ( e ) + z z f ( e ) , AMANUJAN TRIANGLE COMPLEXES 11 which gives (cid:104) δ (cid:12)(cid:12) Ω z ( B ) (cid:105) B E B T = (cid:18) qz qz z z z qz (cid:19) , where B T is the ordered basis f T ( ) , f T (1 2) .The triangles containing e are obtained by adjoining σ v (which gives t ) and (cid:16) π x π (cid:17) v ( x ∈ F q ), giving (cid:16) x (cid:17) t . This yields ( ∂ f ) ( e ) = f ( t ) + qf ( t ) , but for e , e we need towork a little harder, and use (3.7): ( ∂ f ) ( e ) = ( ∂ f ) ((3 2 1) e ) = f ((3 2 1) t ) + (cid:88) x ∈ F q f (cid:16) (3 2 1) (cid:16) x (cid:17) t (cid:17) = 1 qz f ( t ) + f ((3 2 1) t ) + (cid:88) x ∈ F × q f (cid:16) (3 2 1) (cid:16) x (cid:17) (1 2) t (cid:17) = 1 qz f ( t ) + f ((2 3) t ) + (cid:88) x ∈ F × q f (cid:18)(cid:18) − x x (cid:19) (3 2 1) (cid:18) x (cid:19) t (cid:19) = 1 qz f ( t ) + z f ( t ) + (cid:88) x ∈ F × q f ((3 2 1) t ) = 1 z f ( t ) + z f ( t )( ∂ f ) ( e ) = z q f ( t ) + (cid:88) x ∈ F q f (cid:16)(cid:16) x (cid:17) (1 3) t (cid:17) = z q f ( t ) + 1 z f ( t ) , so that (cid:104) ∂ (cid:12)(cid:12) Ω z ( B ) (cid:105) B T B E = q / z z z / q / z , giving (cid:104) ∆ +1 (cid:12)(cid:12) Ω z ( B ) (cid:105) B E = q + 1 qz + qz q z + qz z + z q + 1 qz + qz z q + z z + z q + 1 (cid:104) ∆ − (cid:12)(cid:12) Ω z ( B ) (cid:105) B T = (cid:32) q + 2 qz + qz + q z + z + 1 2 q + 1 (cid:33) . Recalling that λ K = k − q (cid:101) z = 2 (cid:0) q + q + 1 (cid:1) − q (cid:0)(cid:80) z i + z − i (cid:1) , Spec ∆ +1 (cid:12)(cid:12) Ω z ( B ) = (cid:8) λ E , λ E ± (cid:9) := (cid:26) , ( q + 1) ± (cid:113) ( q + 1) + 4 q (2 + (cid:101) z ) (cid:27) = (cid:26) , k ± (cid:113)(cid:0) k (cid:1) − λ K (cid:27) , (3.11)and again Spec ∆ − (cid:12)(cid:12) Ω z ( B ) = (cid:8) λ E ± (cid:9) as we have argued for ∆ − . For ∆ +1 , λ E = 0 is obtained on δ f K (whose f Ew coefficients were computed in (3.8)), and λ E ± are obtained on f E ± = (cid:16) z − + z z (cid:17) q − z + 1) q − q z + q (cid:16) z + z − z − (cid:17) ± ( qz − (cid:112) k − λ K qz (cid:0) z − + 2 z + 1 (cid:1) − z z − z − z ± (cid:16) − z + z z (cid:17) (cid:112) k − λ K T · f E ( ) f E (3 2 1) f E (1 2 3) = 2 q (cid:18) z + 1 z (cid:19) ∂ f T ( ) + (cid:18) q − ± (cid:113) k − λ K (cid:19) ∂ f T (1 2) . AMANUJAN TRIANGLE COMPLEXES 12
Unitary Iwahori-spherical representations.
In general, an irreducible Iwahori-spherical representation is only a subrepresentation of V z . Denote by W z this subrepresenta-tion (there is only one such). Tadic [Tad86] classified the Satake parameters for which therepresentation W z admits a unitary structure, and in [KLW10] the possible z for P GL ( F ) are listed, and a basis for W z ≤ V z is computed explicitly, using results from [Bor76, Zel80].It turns out that a unitary E -spherical W z is of one of the following types:(a) | z i | = 1 for i = 1 , , . In this case W z = V z , and (cid:101) z ∈ [ − , gives λ K ∈ [ k − q, k + 3 q ] and λ E ± ∈ I ± (see (3.11) and (2.3)).(b) z = (cid:0) c − , cq a , cq − a (cid:1) for some | c | = 1 and < a < . Here too W z = V z .(c) z = (cid:16) c √ q , c √ q, c − (cid:17) for some | c | = 1 . In this case W E z is one-dimensional, and spannedby f E − , which is proportional to qf E (3 2 1) − f E (1 2 3) . It corresponds to λ E − = 12 (cid:32) k − (cid:115) k + 8 q + 4 q (cid:18) c √ q + c √ q + c √ q + cq + c − + c (cid:19)(cid:33) = 12 (cid:18) k − (cid:113) q + 8 q √ q (cid:60) ( c ) + 2 q + 16 q (cid:60) ( c ) + 1 + 8 √ q (cid:60) ( c ) (cid:19) = 12 (3 k − ( q + 4 √ q (cid:60) ( c ) + 1)) = k − √ q (cid:60) ( c ) which lies in I (see (2.4)). As f E − is not K -fixed, W K z = 0 .(d) z = (cid:16) c √ q, c √ q , c − (cid:17) for some | c | = 1 . Here W E z = (cid:10) f E , f E + (cid:11) , where f E + is propor-tional to ( q + 1) f E ( ) + (cid:16) c + c √ q (cid:17) (cid:16) f E (3 2 1) + f E (1 2 3) (cid:17) , and λ E + = 2 k + 2 √ q (cid:60) ( c ) simi-larly to the computation in type (c). This time f K is in W E z , and corresponds to λ K = k − q (cid:60) (cid:16) ( q +1) √ q c + c (cid:17) .(e) z = (cid:0) q, , q (cid:1) ; W z is the trivial representation ρ : G → C × , and W E z = W K z are spanned by f K = f E + . Since f K is constant and f E + is a disorientation we have λ K = 0 and λ E + = 3 k (alternatively, use (3.10) and (3.11)).(f) z = (cid:0) ωq, ω, ωq (cid:1) where ω = e ± πi ; W z is the one-dimensional representation ρ ( g ) = ω τ ( g ) ,and W K z = W E z = (cid:10) f K (cid:11) = (cid:10) f E (cid:11) , giving λ K = k .Apart from these there is the Steinberg (Stn) representation z = (cid:0) q , , q (cid:1) . It is not E -spherical,and W T is spanned by f T = qf T ( ) − f T (1 2) , which is always in ker ∂ = ker ∆ − . (In [KLW10]the twisted Steinberg representations z = (cid:0) ωq , ω, ωq (cid:1) are also considered, but they do notcontribute to Ω ∗ as they have no K , E or T -fixed vectors.)Let X = Γ \B be a non-tripartite Ramanujan complex with L (Γ \ G ) ∼ = (cid:76) i W z i , and denoteby N ( t ) the number of W z i of type ( t ) . These are computed in [KLW10] for the tripartite case,and our arguments are similar. By the Ramanujan assumption every Iwahori-spherical W z i iseither tempered, which are the types (a), (c), and (Stn) , or finite-dimensional (types (e), (f)),so that N ( b ) = N ( d ) = 0 . The trivial representation (e) always appears once in L (Γ \ G ) asthe constant functions, so that N ( e ) = 1 . ( † ) Type (f) corresponds to f ∈ L (Γ \ G ) satisfying f (Γ g ) = ( gf ) (Γ) = ω τ ( g ) f (Γ) , which is unique up to scaling, and well defined iff Γ ≤ ker τ , ( † ) This explains why k always appear in Spec ∆ +1 , unlike the graph case, where k ∈ Spec ∆ +0 only forbipartite quotients of B . AMANUJAN TRIANGLE COMPLEXES 13 i.e. X is tripartite. Therefore, N ( f ) = 0 and n = dim Ω ( X ) = (cid:88) i dim W K z i = N ( a ) + N ( e ) + N ( f ) nk ( X ) = (cid:88) i dim W E z i = 3 N ( a ) + N ( c ) + N ( e ) + N ( f ) together imply N ( a ) = n − and N ( c ) = n (cid:0) q + q − (cid:1) + 2 . This is summarized in Table3.1, together with the tripartite case, and this also completes the proof of Theorem 2.3.For completeness, Table 3.1 also shows W T for each type. From nk k = dim Ω ( X ) =2 N ( a ) + N ( c ) + N ( e ) + N ( Stn ) one has N ( Stn ) = (cid:80) i = − ( − i (cid:12)(cid:12) X i (cid:12)(cid:12) = (cid:101) χ ( X ) , the reduced Eulercharacteristic of X .Type W K ∆ +0 e.v. W E ∆ +1 e.v. W T mult . Γ ≤ ker τ mult . Γ (cid:2) ker τ (a) tempered f K k + q (cid:101) z f E , f E ± , k ± (cid:113) k − λ K δ f E ± n − n − (b) f K k + q (cid:101) z f E , f E ± , k ± (cid:113) k − λ K δ f E ± tempered - f E − k − √ q (cid:60) ( c ) δ f E − nq + nq − n +6 nq + nq − n +2 (d) f K k + q (cid:101) z f E , f E + , k + 2 √ q (cid:60) ( c ) δ f E + trivial f K f E + k δ f E + fin. dim. f K k f E (Stn) tempered f T (cid:101) χ ( X ) (cid:101) χ ( X ) Table 3.1.
The representations appearing in L (Γ \ G ) , with the corre-sponding Laplacian eigenvalues, and the multiplicity of appearance in thetripartite and non-tripartite Ramanujan cases.4. Combinatorial expansion
Isoperimetric expansion.
The nontrivial spectrum of ∆ +0 on a non-tripartite Ramanu-jan complex is highly concentrated, lying in a k ± O (cid:0) √ k (cid:1) strip. The nontrivial ∆ +1 -spectrumon -cycles is “almost concentrated”: there are ≈ nq eigenvalues in a k ± O (cid:0) √ k (cid:1) strip, butalso n − eigenvalues at k ± O (1) (and the trivial eigenvalue k ). Nevertheless, havinga concentrated vertex spectrum, and edge spectrum bounded away from zero is enough toprove the Cheeger-type inequality in Theorem 1.3(1): For a partition of the vertices into sets A , A , A of sizes at least ϑn, | X ( A , A , A ) | n | A | | A | | A | ≥ q − q . − C · q ϑ . Remarks. (1) This should be compared to the pseudo-random expectation: X has nk k triangles, so its triangle density is indeed n (cid:0) q + q + 1 (cid:1) ( q + 1) / (cid:0) n (cid:1) ≈ q n .(2) The restriction | A i | ≥ ϑn is essential: If f ( n ) is any sub-linear function, one can take A ⊆ X to be any set of size f ( n ) , A some set containing ∂A = { v | dist ( v, A ) = 1 } ,and A the rest of the vertices. Assuming n is large enough one has | A | , | A | , | A | ≥ f ( n ) , and T ( A , A , A ) = ∅ .(3) Another Cheeger constant for complexes was suggested in [PRT16], and studied in [GS14].However, it is trivial for clique complexes, so we do not address it here.We prove Theorem 1.3(1) as part of Theorem 1.2, which applies to general dimension. AMANUJAN TRIANGLE COMPLEXES 14
Proof of Theorem 1.2.
For f ∈ Ω d − ( X ) defined by f ([ σ σ . . . σ d − ]) = (cid:40) sgn π (cid:12)(cid:12) A π ( d ) (cid:12)(cid:12) ∃ π ∈ Sym { ...d } with σ i ∈ A π ( i ) for 0 ≤ i ≤ d −
10 else , i . e . ∃ k, i (cid:54) = j with σ i , σ j ∈ A k , it is shown in [PRT16, §4.1] that (cid:107) δf (cid:107) = | X ( A , . . . , A d ) | n . ( † ) For f B = P B d − f and f Z = P Z d − f , this gives | X ( A , . . . , A d ) | n = (cid:107) δf (cid:107) = (cid:107) δf Z (cid:107) = (cid:10) ∆ + d − f Z , f Z (cid:11) ≥ λ d − (cid:107) f Z (cid:107) = λ d − (cid:16) (cid:107) f (cid:107) − (cid:107) f B (cid:107) (cid:17) . Denoting K = k · . . . · k d − and E = µ k + . . . + µ d − k d − , we have by [Par17, Thm. 1.3] (cid:107) f (cid:107) = d (cid:88) i =0 (cid:12)(cid:12)(cid:12) X (cid:16) A , . . . , (cid:99) A i , . . . , A d (cid:17)(cid:12)(cid:12)(cid:12) | A i | ≥ d (cid:88) i =0 (cid:20) K n d − (cid:89) j (cid:54) = i | A j | − c d − KE max j (cid:54) = i | A j | (cid:21) | A i | ≥ K n d − (cid:32) d (cid:89) i =0 | A i | (cid:33) − ( d + 1) c d − KE n ≥ K (cid:32) n − d d (cid:89) i =0 | A i | − ( d + 1) c d − E n (cid:33) . Turning to f B , let us denote D = k P B d − − ∆ − d − . Any linear maps T : V → W and S : W → V satisfy (Spec T S ) \ { } = (Spec ST ) \ { } , and thus Spec ∆ − d − (cid:12)(cid:12) B d − = Spec ∆ − d − \ { } = Spec ∆ + d − \ { } = Spec ∆ + d − (cid:12)(cid:12) B d − ⊆ Spec ∆ + d − (cid:12)(cid:12) Z d − ⊆ [ k d − − µ d − , k d − + µ d − ] . Together with ∆ − d − (cid:12)(cid:12) Z d − = 0 this implies (cid:107) D (cid:107) ≤ µ d − , so that (cid:107) f B (cid:107) = (cid:104) P B d − f, f (cid:105) ≤ |(cid:104) D f, f (cid:105)| + (cid:12)(cid:12)(cid:10) ∆ − d − f, f (cid:11)(cid:12)(cid:12) k d − ≤ µ d − k d − (cid:107) f (cid:107) + 1 k d − (cid:107) ∂f (cid:107) . We note that ∂f is supported on ( d − -cells with vertices in distinct blocks of the partition { A i } . For a sequence of sets B , . . . , B (cid:96) , denote by X j ( B , . . . , B (cid:96) ) the set of j -galleries in B , . . . , B (cid:96) , namely, sequences of j -cells τ i ∈ X ( B i , . . . , B i + j ) such that τ i and τ i +1 intersectin a ( j − -cell. To shorten the formulae, we write A [ d ] \{ i,j } for A , . . . , (cid:99) A i , . . . , (cid:99) A j , . . . , A d . Wehave: (cid:107) ∂f (cid:107) = (cid:88) i In this section we use not only the lower bound on the edgespectrum, but the fact that it is concentrated in two narrow stripes, to show a pseudo-randombehavior of -galleries. Theorem 4.1. Let X be an n -vertex tripartite triangle complex with vertex and edge degrees k and k , such that Spec ∆ +0 (cid:12)(cid:12) Z ⊆ [ k − µ , k + µ ] ∪ (cid:8) k (cid:9) and Spec ∆ +1 (cid:12)(cid:12) Z ⊆ [ k − µ , k + µ ] ∪ [2 k − µ , k + µ ] ∪ { k } . If A, B, C, D are disjoint sets of vertices of sizes a, b, c, d , respectively, and each of A ∪ D , B and C is contained in a different block of the three-partition of X (see Figure 4.1), then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( A, B, C, D ) (cid:12)(cid:12) − k k abcd n (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ k √ abcdk n k (cid:16) √ ab + √ cd (cid:17) n + µ + (cid:20) k µ k + ( k + µ ) µ (cid:21) √ abcd (cid:114)(cid:16) k √ ab n + µ (cid:17) (cid:16) k √ cd n + µ (cid:17) . (4.3) Figure 4.1. A -gallery through A , B , C , D in a tripartite trianglecomplex. AMANUJAN TRIANGLE COMPLEXES 16 It follows that if a, b, c, d ≤ ϑn (where ϑ ≤ ) then the l.h.s. in (4.3) is bounded by (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( A, B, C, D ) (cid:12)(cid:12) − k k abcd n (cid:12)(cid:12)(cid:12)(cid:12) ≤ ϑ (cid:18) ϑ + 4 µ k (cid:19) ( k µ + k µ ) k n, and for Ramanujan complexes k = 2 (cid:0) q + q + 1 (cid:1) , k = q + 1 , µ = 6 q and µ = 2 √ q , whichgives Theorem 1.3(2). The main term in (4.3) agrees with the pseudo-random expectation:given vertices α, β, γ, δ in A, B, C, D , respectively, the probability that βγ is an edge in X is k n , and the k triangles which contain it have their third vertex in the block containing A ∪ D .The probability that α and δ are two of these is k ( k − n / ( n / − , so that E (cid:0)(cid:12)(cid:12) X ( A, B, C, D ) (cid:12)(cid:12)(cid:1) = k n · k ( k − n / ( n / − · abcd ≈ k k abcd n .We shall need a c -partite version of the expander mixing lemma, where we say that a k -regulargraph ( V, E ) on n vertices is c -partite if V = V (cid:116) . . . (cid:116) V c − with | V i | = nc so that E ( V i , V i ) = ∅ and | E ( v, V j ) | = kc − for v ∈ V i and j (cid:54) = i . The functions f j ( V (cid:96) ) ≡ exp (cid:16) πij(cid:96)c (cid:17) / √ n areorthonormal eigenfunctions of ∆ +0 with corresponding eigenvalues λ j = (cid:40) j = 0 (cid:16) cc − (cid:17) k < j < c, (4.4)and we call { λ , . . . λ c − } the partite spectrum . Lemma 4.2. If the non-partite spectrum of a c -partite k -regular graph on n vertices is con-tained in [ k − µ, k + µ ] , and A ⊆ V i , B ⊆ V j for i (cid:54) = j , then (cid:12)(cid:12)(cid:12)(cid:12) | E ( A, B ) | − ck | A | | B | ( c − n (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ (cid:112) | A | | B | . (4.5) Proof. Assuming that A ⊆ V and B ⊆ V , and denoting by P W the orthogonal projectionon W = (cid:104) f , . . . , f c − (cid:105) ⊥ , we have | E ( A, B ) | = (cid:10)(cid:0) kI − ∆ +0 (cid:1) A , B (cid:11) = c − (cid:88) j =0 ( k − λ j ) (cid:104) A , f j (cid:105) (cid:104) B , f j (cid:105) + (cid:10)(cid:0) kI − ∆ +0 (cid:1) P W A , B (cid:11) = | A | | B | n c − (cid:88) j =0 ( k − λ j ) exp (cid:18) πijc (cid:19) + (cid:10)(cid:0) kI − ∆ +0 (cid:1) P W A , B (cid:11) , and (4.5) follows by (4.4) and (cid:13)(cid:13)(cid:0) kI − ∆ +0 (cid:1) (cid:12)(cid:12) W (cid:13)(cid:13) ≤ µ . (cid:3) Proof of Theorem 4.1. Denote by U + the span of the ∆ +1 -eigenforms with eigenvalues in [ k − µ , k + µ ] ∪ [2 k − µ , k + µ ] , and by η a normalized k -eigenform for ∆ +1 , so that Ω ( X ) = B ⊕ U + ⊕ (cid:104) η (cid:105) . Denoting p ( x ) = ( x − k ) ( x − k ) , p (cid:0) ∆ +1 (cid:1) acts on B ⊕ (cid:104) η (cid:105) asthe scalar k , and (cid:13)(cid:13) p (cid:0) ∆ +1 (cid:1) (cid:12)(cid:12) U + (cid:13)(cid:13) ≤ max (cid:110) | p ( λ ) | (cid:12)(cid:12)(cid:12) λ ∈ [ k − µ ,k + µ ] ∪ [2 k − µ , k + µ ] (cid:111) = ( k + µ ) µ . Say that two directed edges are neighbors if they have a common origin or a common terminus,and their union (as a cell) is in X . We denote this by e ∼ e (cid:48) , and define A : Ω ( X ) → Ω ( X ) AMANUJAN TRIANGLE COMPLEXES 17 by ( A f ) ( e ) = (cid:80) e (cid:48) ∼ e f ( e (cid:48) ) . The upper Laplacian satisfies ∆ +1 = k · I − A (see [Par17]), andit follows that p (cid:0) ∆ +1 (cid:1) = A + k A . Define AB ∈ Ω ( X ) by AB ( vw ) = v ∈ A, w ∈ B − v ∈ B, w ∈ A otherwise and similarly CD . We claim that (cid:12)(cid:12) X ( A, B, C, D ) (cid:12)(cid:12) = (cid:10) p (cid:0) ∆ +1 (cid:1) AB , CD (cid:11) . (4.6)Indeed, edges in E ( A, B ) have no neighbors in E ( C, D ) , so that (cid:104)A AB , CD (cid:105) = 0 , and A isself-adjoint (since ∆ +1 is), giving (cid:10) p (cid:0) ∆ +1 (cid:1) AB , CD (cid:11) = (cid:104)A AB , A CD (cid:105) = (cid:88) e,e (cid:48) ,e (cid:48)(cid:48)∈ X e (cid:48)∼ e ∼ e (cid:48)(cid:48) AB ( e (cid:48) ) CD ( e (cid:48)(cid:48) ) . (4.7)The nonzero terms in this sum come from edges e which have neighbors e (cid:48) ∈ E ( A, B ) and e (cid:48)(cid:48) ∈ E ( C, D ) , and it follows that e ∈ E ( B, C ) . Thus, ( e (cid:48) ∪ e, e ∪ e (cid:48)(cid:48) ) is a -gallery in X ( A, B, C, D ) , and observing that AB ( e (cid:48) ) = CD ( e (cid:48)(cid:48) ) (= ± it contributes one to (4.7).On the other hand, for every gallery ( t, t (cid:48) ) ∈ X ( A, B, C, D ) , the edges e (cid:48) = t \ C , e = t ∩ t (cid:48) , e (cid:48)(cid:48) = t \ B form such a triplet, and we obtain (4.6). On the spectral side, (cid:10) p (cid:0) ∆ +1 (cid:1) AB , CD (cid:11) = 2 k (cid:10) P B ⊕(cid:104) η (cid:105) AB , CD (cid:11) + (cid:10) p (cid:0) ∆ (cid:1) P U + AB , CD (cid:11) , and the last term is bounded by (cid:13)(cid:13) p (cid:0) ∆ (cid:1) (cid:12)(cid:12) U + (cid:13)(cid:13) (cid:107) AB (cid:107) (cid:107) CD (cid:107) ≤ ( k + µ ) µ (cid:112) E AB E CD , (4.8)where E ST := | E ( S, T ) | . As η has constant sign on V → V → V → V , k (cid:10) P (cid:104) η (cid:105) AB , CD (cid:11) = 2 k (cid:104) AB , η (cid:105) (cid:104) η, CD (cid:105) = 4 k E AB E CD k n , (4.9)and we are left to analyze P B AB . As in the non-tripartite case, one has Spec ∆ − (cid:12)(cid:12) B =Spec ∆ (cid:12)(cid:12) B , but now the latter comprises not only eigenvalues in [ k − µ , k + µ ] , but also k (twice, see Theorem 2.3). If ω = exp (cid:0) πi (cid:1) and ξ ( vw ) = (cid:112) / k n v ∈ V , w ∈ V − (cid:112) / k n w ∈ V , v ∈ V ω (cid:112) / k n v ∈ V , w ∈ V − ω (cid:112) / k n w ∈ V , v ∈ V ω (cid:112) / k n v ∈ V , w ∈ V − ω (cid:112) / k n w ∈ V , v ∈ V , then (cid:8) ξ, ξ (cid:9) is an orthonormal basis for the k -eigenspace of ∆ − . Denote by U − the spacespanned by the ∆ − -eigenforms with eigenvalue in [ k − µ , k + µ ] . By the action of eachsummand in D (cid:48) := k P B + k P (cid:104) ξ,ξ (cid:105) − ∆ − on each of the terms in the orthogonal decomposition Ω ( X ) = Z ⊕ U − ⊕ (cid:10) ξ, ξ (cid:11) we seethat (cid:107) D (cid:48) (cid:107) ≤ µ . Due to the fact that ∂ AB and ∂ CD are supported on different vertices, (cid:10) ∆ − AB , CD (cid:11) vanishes, and together with (cid:68) P (cid:104) ξ,ξ (cid:105) AB , CD (cid:69) = 2 (cid:60) ( (cid:104) AB , ξ (cid:105) (cid:104) ξ, CD (cid:105) ) = − E AB E CD k n AMANUJAN TRIANGLE COMPLEXES 18 (and P B = k ∆ − − P (cid:104) ξ,ξ (cid:105) + k D (cid:48) ) this gives (cid:12)(cid:12)(cid:12)(cid:12) k (cid:104) P B AB , CD (cid:105) − k E AB E CD k n (cid:12)(cid:12)(cid:12)(cid:12) ≤ k k (cid:104) D (cid:48) AB , CD (cid:105) ≤ µ k √ E AB E CD k . Combining this with (4.8) and (4.9) we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( A, B, C, D ) (cid:12)(cid:12) − k E AB E CD k n (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) µ k k + ( k + µ ) µ (cid:19) (cid:112) E AB E CD . This estimates (cid:12)(cid:12) X ( A, B, C, D ) (cid:12)(cid:12) in terms of E AB and E CD . To have an estimate in terms of a, b, c and d we use Lemma 4.2, which gives (cid:12)(cid:12) E AB − k ab n (cid:12)(cid:12) ≤ µ √ ab and similarly for E CD ,and the theorem follows. (cid:3) Various applications of a triangle mixing lemma can be adjusted to use our gallery mixinglemma. We demonstrate this below for chromatic numbers, and other examples are Gromov’soverlap property, along the lines of [FGL + 12, Par17], and the crossing numbers of complexes,as discussed in [GW13, §8.1]. Nevertheless, the question of triangle pseudorandomness re-mains interesting, and should give better results if it does hold. The fact that most of thespectrum is concentrated in the strip I (see Theorem 2.3) gives hope that this can be doneby analyzing the combinatorics of eigenforms which occur in the principal series (type (a) in§3.3), and showing that their contribution is negligible.4.3. Chromatic Number. As an application of Theorem 4.1 we prove Theorem 1.3(3),which bounds the chromatic number of non-tripartite Ramanujan complexes. Proof of Theorem 1.3(3). Write X = Γ \B and let (cid:98) Γ = Γ ∩ ker τ . This is a normal sub-group of Γ of index three, and (cid:98) X := (cid:98) Γ \B π (cid:16) X is a tripartite three-cover. If the chro-matic number of X is χ , we can find a set N ⊆ X of size nχ with T ( N, N, N ) = ∅ .Let N i = (cid:110) v ∈ (cid:98) X (cid:12)(cid:12)(cid:12) π ( v ) ∈ N, τ ( v ) = i (cid:111) , and take A = N , B = N , C = N and D ⊆ (cid:110) v ∈ (cid:98) X (cid:12)(cid:12)(cid:12) τ ( v ) = 0 (cid:111) \ N such that | D | = n . Since the set T ( N, N, N ) is empty, X ( A, B, C, D ) is empty as well. Therefore the l.h.s. of (4.3) reads q n χ = q abcdn ≤ k k abcd n ) .Assume to the contrary that χ < √ q . Then the r.h.s. of (4.3) is bounded by nq . χ . √ (cid:18) (cid:16) 264 + 255 √ (cid:17)(cid:19) < nq . χ . · √ , so that Theorem 4.1 implies √ q < √ χ . , which contradicts the assumption. (cid:3) References [AC88] N. Alon and F.R.K. Chung, Explicit construction of linear sized tolerant networks , DiscreteMathematics (1988), no. 1-3, 15–19.[AM85] N. Alon and V.D. Milman, λ , isoperimetric inequalities for graphs, and superconcentrators ,Journal of Combinatorial Theory, Series B (1985), no. 1, 73–88.[Bor76] A. Borel, Admissible representations of a semi-simple group over a local field with vectorsfixed under an Iwahori subgroup , Inventiones mathematicae (1976), no. 1, 233–259.[Cas74] W. Casselman, On a p -adic vanishing theorem of Garland , Bulletin of the American Math-ematical Society (1974), no. 5, 1001–1004.[Cas80] , The unramified principal series of p -adic groups. I. The spherical function , Com-positio Mathematica (1980), no. 3, 387–406. AMANUJAN TRIANGLE COMPLEXES 19 [CHH88] M. Cowling, U. Haagerup, and R. Howe, Almost L matrix coefficients. , J. Reine Angew.Math. (1988), 97–110.[CS02] D.I. Cartwright and T. Steger, Elementary symmetric polynomials in numbers of modulus1 , Canad. J. Math. (2002), no. 2, 239–262. MR 1892996 (2003h:05188)[CSŻ03] D.I. Cartwright, P. Solé, and A. Żuk, Ramanujan geometries of type ˜ A n , Discrete mathe-matics (2003), no. 1, 35–43.[Eck44] B. Eckmann, Harmonische funktionen und randwertaufgaben in einem komplex , Commen-tarii Mathematici Helvetici (1944), no. 1, 240–255.[EGL15] S. Evra, K. Golubev, and A. Lubotzky, Mixing properties and the chromatic number ofRamanujan complexes , International Mathematics Research Notices (2015).[FGL + 12] J. Fox, M. Gromov, V. Lafforgue, A. Naor, and J. Pach, Overlap properties of geometricexpanders , J. Reine Angew. Math. (2012), 49–83.[Fir16] U. A. First, The Ramanujan property for simplicial complexes , arXiv preprintarXiv:1605.02664 (2016).[FP87] J. Friedman and N. Pippenger, Expanding graphs contain all small trees , Combinatorica (1987), no. 1, 71–76.[Gar73] H. Garland, p-adic curvature and the cohomology of discrete subgroups of p-adic groups ,The Annals of Mathematics (1973), no. 3, 375–423.[GS14] A. Gundert and M. Szedlák, Higher dimensional Cheeger inequalities , Annual Symposiumon Computational Geometry (New York, NY, USA), SOCG’14, ACM, 2014, pp. 181:181–181:188.[GW13] A. Gundert and U. Wagner, On expansion and spectral properties of simplicial complexes ,Ph.D. thesis, ETH Zurich, Switzerland, 2013, Diss. ETH No. 21600 of Anna Gundert.[HLW06] S. Hoory, N. Linial, and A. Wigderson, Expander graphs and their applications , Bulletin ofthe American Mathematical Society (2006), no. 4, 439–562.[Hof70] Alan J. Hoffman, On eigenvalues and colorings of graphs , Graph Theory and its Applica-tions (Proc. Advanced Sem., Math. Research Center, Univ. of Wisconsin, Madison, Wis.,1969), Academic Press, New York, 1970, pp. 79–91.[KLW10] M.H. Kang, W.C.W. Li, and C.J. Wang, The zeta functions of complexes from PGL(3) : arepresentation-theoretic approach , Israel Journal of Mathematics (2010), no. 1, 335–348.[Li04] W.C.W. Li, Ramanujan hypergraphs , Geometric and Functional Analysis (2004), no. 2,380–399.[LM07] A. Lubotzky and R. Meshulam, A Moore bound for simplicial complexes , Bulletin of theLondon Mathematical Society (2007), no. 3, 353–358.[LPS88] A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs , Combinatorica (1988), no. 3,261–277.[LSV05a] A. Lubotzky, B. Samuels, and U. Vishne, Ramanujan complexes of type ˜ A d , Israel Journalof Mathematics (2005), no. 1, 267–299.[LSV05b] , Explicit constructions of Ramanujan complexes of type ˜ A d , Eur. J. Comb. (2005), no. 6, 965–993.[Lub94] A. Lubotzky, Discrete groups, expanding graphs and invariant measures , ModernBirkhäuser Classics, Birkhäuser Verlag, Basel, 1994, With an appendix by Jonathan D.Rogawski.[Lub12] , Expander graphs in pure and applied mathematics , Bull. Amer. Math. Soc (2012), 113–162.[Lub14] , Ramanujan complexes and high dimensional expanders , Japanese Journal of Math-ematics (2014), no. 2, 137–169.[Mac79] I.G. Macdonald, Symmetric functions and Hall polynomials , Clarendon Press Oxford, 1979.[Mar88] G.A. Margulis, Explicit group-theoretical constructions of combinatorial schemes and theirapplication to the design of expanders and concentrators , Problemy Peredachi Informatsii (1988), no. 1, 51–60. AMANUJAN TRIANGLE COMPLEXES 20 [Pap08] M. Papikian, On eigenvalues of p -adic curvature , Manuscripta mathematica (2008),no. 3, 397–410.[Par17] O. Parzanchevski, Mixing in high-dimensional expanders , Combinatorics, Probability andComputing (2017), online, doi:10.1017/S0963548317000116.[PR17] Ori Parzanchevski and Ron Rosenthal, Simplicial complexes: Spectrum, homology andrandom walks , Random Structures & Algorithms (2017), no. 2, 225–261.[PRT16] O. Parzanchevski, R. Rosenthal, and R.J. Tessler, Isoperimetric inequalities in simplicialcomplexes , Combinatorica (2016), no. 2, 195–227.[Sar90] P. Sarnak, Some applications of modular forms , vol. 99, Cambridge University Press, 1990.[Sar07] A. Sarveniazi, Explicit construction of a Ramanujan ( n , n , . . . , n d − ) -regular hypergraph ,Duke Mathematical Journal (2007), no. 1, 141–171.[Tad86] M. Tadic, Classification of unitary representations in irreducible representations of generallinear group (non-archimedean case) , Annales scientifiques de l’Ecole normale supérieure (1986), no. 3, 335–382.[Zel80] A.V. Zelevinsky, Induced representations of reductive p-adic groups II. On irreducible repre-sentations of GL(n) , Annales Scientifiques de l’École Normale Supérieure (1980), no. 2,165–210. Einstein Institute of MathematicsThe Hebrew University of JerusalemJerusalem, 91904, Israel E-mail: [email protected] School of MathematicsInstitute for Advanced StudyPrinceton, NJ 08540, USA