High Dimensional Expanders From Ramanujan Graphs to Ramanujan Complexes Alex Lubotzky Einstein Institute of Mathematics, Hebrew University Jerusalem, ISRAEL A. Lubotzky (Hebrew University) 1 / 17
1 Ramanujan graphs X - a connected k -regular graph with A = A X - adjacency matrix A v,u = # edges between u and v Eigenvalues: k = λ 0 ≥ λ 1 ≥ . . . ≥ λ n − 1 ≥ − k Definition X is called Ramanujan graph if for every λ eigenvalue of A , either | λ | = k or √ | λ | ≤ 2 k − 1 � � � � i.e. for all non-trivial e.v. λ , λ ∈ Spec A . L 2 ( T k ) • The eigenvalues control the rate of convergence of the random walk to uniform distribution • Ramanujan ⇒ a) fastest rate b) best expanders (Alon-Boppana) A. Lubotzky (Hebrew University) 2 / 17
Explicit construction Let p � = q primes p ≡ q ≡ 1 (mod 4) Jacobi Theorem: � � � � 3 � � � ( x 0 , x 1 , x 2 , x 3 ) ∈ Z 4 x 2 r 4 ( n ) := # i = n = 8 d � � i =0 d | n 4 ∤ d Thus r 4 ( p ) = 8 ( p + 1) For our p , p ≡ 1 (mod 4) so one x i is odd and three are even . � α = ( x 0 , x 1 , x 2 , x 3 ) ∈ Z 4 � � � x 2 � Let S = i = p, x 0 > 0 , odd ∴ | S | = p + 1 Think of α as integral quaternion α = x 0 + x 1 i + x 2 j + x 3 k so α ∈ S ⇒ α ∈ S and � α � = αα = p . A. Lubotzky (Hebrew University) 3 / 17
As q ≡ 1 (4) take ε ∈ F q with ε 2 = − 1 For α ∈ S , let: � x 0 + εx 1 � x 2 + εx 3 α = � ∈ PGL 2 ( F q ) − x 2 + εx 3 x 0 − εx 1 Theorem (Lubotzky-Phillips-Sarnak 1986) α | α ∈ S � and X p,q = Cay ( H, { � Let H = � � α } ) . Then X p,q is a ( p + 1) -regular Ramanujan graph. 1 � � p If = − 1 , i.e. p not quadratic residue mod q , then 2 q H = PGL 2 ( F q ) and X p,q is bi-partite. Otherwise, H = PSL 2 ( F q ) and X p,q is not. 3 Note • | λ | = p + 1 or | λ | ≤ 2 √ p (Riemann hypothesis over finite fields). • Zeta functions approach (Ihara, Sunada, Hashimoto, ...) . A. Lubotzky (Hebrew University) 4 / 17
Where do X p,q come from? Let F = Q p or F p (( t )) O = Z p or F p [[ t ]] - the ring of integers. G = PGL 2 ( F ) , K = PGL 2 ( O ) - maximal compact in G . G/K = ( p + 1) -regular tree = The Bruhat-Tits tree. If Γ ≤ G a discrete cocompact subgroup (a lattice) then Γ \ G/K = Γ \ T = a finite ( p + 1) -regular graph!! A. Lubotzky (Hebrew University) 5 / 17
Theorem Γ \ G/K = Γ \ T is Ramanujan iff every infinite dimensional irreducible spherical subrepresentation of L 2 (Γ \ G ) (as G -rep) is tempered. spherical ≡ has non-zero K -fixed point tempered ≡ matrix coef’s are in L 2+ ε ≡ weakly contained in L 2 ( G ) So the combinatorial property of being Ramanujan is equivalent to a representation theoretic statement. The latter one is actually number theoretic (Satake). A. Lubotzky (Hebrew University) 6 / 17
Theorem (Deligne) If Γ is an arithmetic lattice of PGL 2 ( Q p ) and Γ( I ) a congruence subgroup then every irr. ∞ -dim spherical subrepresentation of L 2 (Γ( I ) \ G ) is tempered. Corollary Γ( I ) \ T = Γ( I ) \ G/K is a Ramanujan graph. The explicit expanders above are obtained from an especially nice Γ (Hamiltonian quaternions) • Similar result by Drinfeld in positive characteristic • Similar construction by Morgenstern, ∀ k = p α + 1 Many applications � p � E.g. Let X = X p,q , = 1 then X has large girth and large chromatic number. q Compare: Erd¨ os, Lov´ asz A. Lubotzky (Hebrew University) 7 / 17
2 Ramanujan Complexes � The generalization of T = PGL 2 ( F ) K is the Bruhat-Tits building � B d ( F ) = G/K = PGL d ( F ) PGL d ( O ) , a ( d − 1) -dim contractible simplicial complex. � The vertices of the building B d ( F ) come with “colors” ν ( gK ) ∈ Z d Z , ν ( gK ) = val p (det( g ))(mod d ) . Colored adjacency operators (Hecke operators) A i : L 2 ( B d ( F )) → L 2 ( B d ( F )) ( 1 ≤ i ≤ d − 1 ) For f : B d ( F ) → C , � ( A i f ) ( x ) = f ( y ) y ∼ x ν ( y ) − ν ( x )= i d − 1 � So: Adj = A i . i =1 A. Lubotzky (Hebrew University) 8 / 17
The A i ’s are normal commutating operators (but not self adjoint; in fact A ∗ i = A d − i ), so can be diagonalized simultaneously Σ d := Spec { A 1 , . . . , A d − 1 } ⊆ C d − 1 . Definition � A finite quotient Γ B d ( F ) , Γ cocompact discrete subgroup is a Ramanujan complex if every nontrivial simultaneous eigenvalue ( λ ) = ( λ 1 , . . . , λ d − 1 ) of ( A 1 , . . . , A d − 1 ) acting on L 2 � � � Γ B d ( F ) is in Σ d . Theorem (Li) (` a la Alon-Boppana) � If a sequence of quotients X i = Γ i B d ( F ) has injective radius → ∞ , then � Σ d ⊆ spec X i ( A 1 , . . . , A d − 1 ) . A. Lubotzky (Hebrew University) 9 / 17
Theorem (Lubotzky-Samuels-Vishne 2005) � Γ B d ( F ) is Ramanujan iff every ∞ -dim irreducible spherical subrepresentation of L 2 � � � Γ PGL d ( F ) is tempered. Theorem (Lafforgue 2002) If char F > 0 and Γ an arithmetic subgroup of PGL d ( F ) , and Γ ( I ) congruence subgroup then (under some restrictions) every ........ subrepresentation of L 2 � � � Γ ( I ) PGL d ( F ) is tempered. Corollary � Γ ( I ) B d ( F ) are Ramanujan complexes. • Lubotzky-Samuels-Vishne used it for explicit construction • See also Winnie Li, Sarveniazi A. Lubotzky (Hebrew University) 10 / 17
The constructions are quite complicated, but it has turned out to be a good investment A. Lubotzky (Hebrew University) 11 / 17
3 Overlapping properties Theorem (Boros-F¨ uredi ’84) � n � Given a set P ⊆ R 2 , with | P | = n , ∃ z ∈ R 2 which is covered by ( 2 9 − o (1)) of � n � 3 the triangles determined by P . 3 Remark: 2 9 is optimal. Theorem (Barany) ∀ d ≥ 2 , ∃ c d > 0 s.t. ∀ P ⊂ R d with | P | = n , ∃ z ∈ R d which is covered by at least � n � c d of the d -simplices determined by P . d +1 A. Lubotzky (Hebrew University) 12 / 17
Gromov proved the following remarkable result; but first a definition: Definition (Gromov) A simplical complex X of dimension d has ε -geometric (resp. topological ) overlapping property if for every f : X (0) → R d and every affine (resp. continuous) extension f : X → R d , there exists a point z ∈ R d which is covered by ε · | X ( d ) | of the d -cells of X . A family of s.c.’s of dim d are geometric (resp. topological) expanders if all have it with the same ε . Remark: For d = 1 , EXPANDERS ⇒ TOP. OVERLAPPING. Theorem (Boros-Furedi for d = 2 , Barany for all d -80’s) The complete d -dim s.c. on n vertices ( n → ∞ ) geometric expanders. A. Lubotzky (Hebrew University) 13 / 17
Theorem (Gromov 2010) They are also topological expanders! Think even about d = 2 to see how this special case is non-trivial and even counter intuitive. He also claimed it for spherical building (See Lubotzky-Meshulam-Mozes). Question (Gromov) What about s.c. of bounded degree? Theorem (Fox-Gromov-Lafforgue-Naor-Pach 2013) The Ramanujan complexes of dim d , when q >> 0 , are geometric expanders. What about topological expanders? A. Lubotzky (Hebrew University) 14 / 17
Theorem (Kaufman-Kazhdan-Lubotzky 2016 for d = 2 and Evra-Kaufman 2017 for all d ) Fix q = q ( d ) >> 0 , the ( d − 1) -skeletons of the d -dimensional Ramanujan complexes are ( d − 1) -dim topological expanders. On the connection to Linial-Meshulam high dim expanders and some results on Random - see Lubotzky-Meshulam 2014 for d = 2 and Lubotzky-Luria-Rosenthal 2016 for all d . A. Lubotzky (Hebrew University) 15 / 17
Outline of proof Coboundary expanders X - s.c. of dim d , X ( i ) = i -cells, C i ( X, F 2 ) = F 2 -cochains, with “natural” norm. δ = δ i : C i → C i +1 coboundary map, B i = Image( δ i ) . � � � δ i α � � � � � α ∈ C i \ B i E i = sup for i = 0 , ..., d − 1 � [ α ] � E 0 = normalized Cheeger constant. 1 E i > 0 ⇔ H i ( X, F 2 ) = 0 2 Theorem (Gromov): E -coboundary expansion ⇒ topological expanders 3 Linial-Meshulam: Random s.c.’s. 4 Kaufman-Lubotzky: Property testing. 5 A. Lubotzky (Hebrew University) 16 / 17
Ramanujan complexes are in general not coboundary expanders. 1 ( H i ( X, R ) = 0 by Garland, but H i ( X, F 2 ) can be � = 0. a la Serre, H 1 ( X, F 2 ) = 0 for infinitely many X ). By CSP ` Extension by Kaufman & Wagner of Gromov, in case of H i � = 0 (systolic 2 inequalities). Main technical result: q ≫ 0 , ∃ ε 0 , ε 1 , ε 2 , η 0 , η 1 , η 2 s.t. if α ∈ C i “locally 3 minimal” cochain with � α � ≤ η i , then � δ i α � ≥ ε i � α � . 3 -dim structure plays an essential role even though we eventually ignore it. 4 A. Lubotzky (Hebrew University) 17 / 17
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