Cayley graphs and Expanders

Transcription

1 Cayley graphs and Expanders Peter Dixon April 29, 2016

2 Cayley graphs Definition (Cayley graph) Given a group H and a set S H, the Cayley graph C(H, S) is the graph with vertex set H and edge g, h if there is some s S such that gṡ = h. We will only consider sets S that contain s 1 whenever they contain s, as this guarantees a nice set of properties for C(H, S).

3 Properties of C(H, S) when S is nice 1. Undirected 2. S regular 3. Connected iff S generates H 4. Vertex-transitive - given any pair of vertices v and w, there is an automorphism of C(H, S) that sends v to w. That automorphism is x vw 1 x

4 Families of expanders Definition (Family of expander graphs) is a sequence of d regular graphs G i (i N) increasing in size with i where the Cheeger constant of all G i is at least ɛ > 0. Definition (Made into a family of expanders) A family of graphs H i (i N) can be made into a family of expanders, if there is a generating set S i of size d in each H i such that C(H i, S i ) is a family of expanders. Lubotzky showed that a family consisting of most simple groups can be made into a family of expanders WHAAAAAT

5 Eigenvalues, Spectral gap Definition (λ(h, S)) λ(h, S) is shorthand for the second-largest eigenvalue of the normalized adjacency matrix of C(H, S). The largest eigenvalue is 1, cos normalized. Definition (Spectral gap) The spectral gap g(h, S) is 1 λ(h, S) A small λ/large spectral gap means the graph is a good expander.

6 Bounding λ(h, S) Theorem (Upper bound (Alon, Roichman 94)) Let H be a group and let S be a subset chosen uniformly at random from H of size 100log H. Then λ(h, S) < 1/2 with probability at least 1/2. Theorem (Lower bound, Abelian only) Let H be an Abelian group and let S be a generating set such that λ(h, S) 1/2. Then S log H /3

7 Semidirect Product Definition (Group action) Given groups A, B, we say B acts on A if there is a group homomorphism φ that maps B to Aut(A). We use φ b for φ(b), the automorphism of A associated with b. Definition (A B) Let A be a group and B a group which acts on A. The semidirect product of A and B is the group with elements (a, b) and operation (a 1, b 1 ) (a 2, b 2 ) = (a 1 φ b1 (a 2 ), b 1 b 2 ).

8 Replacement and Zig-zag Products Definition (Replacement Product) Given a m-regular graph G and an order m graph H, the replacement product GRH is formed by replacing each vertex of G with a copy of H. To be more precise, it is the graph with vertex set G H and two kinds of edges: 1) whenever < h j, h k > E(H), < (g i, h j ), (g i, h k ) > E(GRH) 2) whenever (g i, g j ) E(G), there is exactly one h k s.t. < (g i, h k ), (g j, h k ) > E(GRH)

9 Replacement and Zig-zag Products 2 Definition (Zig-zag Product) The zig-zag product GZH is formed by taking the replacement graph GRH and adding additional edges from (g 1, h 1 ) to (g 2, h 2 ) whenever there is a path of length 3 from (g 1, h 1 ) to (g 2, h 2 ) that alternates between type 1 and type 2 edges.

10 Connecting graph and group products Given two groups A, B with generating sets S A, S B, under certain conditions the replacement product of the Cayley graphs C(A, S A ), C(B, S B ) is the Cayley graph of the semidirect product of A and B, C(A B, S), for the right S. Specifically, when 1. B = S A 2. There is some x A such that S A is the orbit of x under the action of B 3. S = {(1, s) s S B } {x, 1}

11 Proof of that last thing Proof: S generates A B. Because S A is the orbit of x under B, there is some b such that φ b (x) = a for any a S A. So we can write (a, 1) as (1, b) (x, 1) (1, b 1 ). Now, to write any (a, b) A B, we do (a 1, 1) (a n, 1) (1, b 1 ) (1, b m ), where a 1,... a n are the elements of S A that generate a and similarly for b 1... b m.

12 Proof of that last thing pt 2 Proof: C(A B, S) = C(A, S A )RC(B, S B ). For a fixed a i, if < b 1, b 2 > is an edge in C(B, S B ) then s b s.t. b 1 s b = b 2, so (a i, b 1 ) (1, s b ) = (a i φ b1 (1), b 1 s b ) = (a i, b 2 ), so < (a i, b 1 ), (a i, b 2 ) > is an edge of C(A B, S). If < a 1, a 2 > is an edge of C(A, S A ) then s a s.t. a 1 s a = a 2. Because S A = orb B (x), b B s.t. φ b (x) = s a. So, (a 1, b) (x, 1) = (a 1 φ b (x), b 1) = (a 1 s a, b) = (a 2, b).

13 Theorem (For a different choice of S, we get the zig-zag product) If we take S = {(1, s 1 ) (x, 1) (1, s 2 ) s 1, s 2 B}, then this gives us all the zig-zag edges. Proof. It was in the paper, so it has to be true.

14 so uh peter... who cares? The zig-zag product has the neat property of almost-preserving expandiness. Theorem (Zig-Zag Theorem (Reingold, Vadhan, Wigderson 02)) Let G be a (n, m, α) graph and H a (m, d, β) graph. Then GZH is an (nm, d 2, φ(α, β)) graph where φ satisfies 1. If α < 1 and β < 1 then φ(α, β) < 1 2. φ(α, β) α + β 3. φ(α, β) 1 (1 β 2 )(1 α 2 )/2

15 peter i still don t care Putting these two together lets us take a few Cayley expander graphs and make an infinite family of Cayley expanders: Theorem (Meshulam, Wigderson 02) There exists a group H 1, a sequence of primes p i, and a sequence of generating sets U i for H i such that λ(h n, U n ) 1/2 and U n log (n/2) H n Theorem (Rozenman, Shalev, Wigderson 04) Let H be a group where every h H is a commutator. Let U be an expanding generating set for H with second eigenvalue at most 1/4 and suppose U < d/10 6. Then λ(h d, U ( d)) < 1/2.