a. vdovina newcastle university oxfordpeople.maths.ox.ac.uk/drutu/conference/vdovinaslides.pdf ·...

What are expanders? Spectral properties of expanders New examples of expander families (joint work with N.Peyerimhoff) On SQ-universality of small cancellation

Cayley graph expanders, buildings and Beauville structures

A. VdovinaNewcastle University

Oxford11 November 2010




Outline

What are expanders?

Spectral properties of expanders

New examples of expander families (joint work with N.Peyerimhoff)

On SQ-universality of small cancellation groups (joint work with Martin Edjvet)

More about methods

New girth and finite simple groups (joint work with R.Blok and C.Hoffman)

Beauville surfaces (joint work with N.Barker,N.Boston,N.Peyerimhoff)




Part 1: What are expanders?Very recommendable survey article:S. Hoory, N. Linial, A. Wigderson: Expander graphs and their applications,Bulletin of the American Mathematical Society 43, 2006, pp. 439-561.




(Discrete) Cheeger constant and regular tree

Assume that X = graph Gwith V = vertex set and E = set of edges. Then

h(G) = inffiniteA⊂V

|∂A|min(|A|, |V\A|) ,

where ∂A = all edges connecting a vertex of A with a vertex of V\A.




(Discrete) Cheeger constant and regular tree

Assume that X = graph Gwith V = vertex set and E = set of edges. Then

h(G) = inffiniteA⊂V

|∂A|min(|A|, |V\A|) ,

where ∂A = all edges connecting a vertex of A with a vertex of V\A.

Example: G = Tp (p-regular infinite tree).

A

T3




Cheeger constant of regular graphs

We have h(Tp) = p− 2. Explanation, why h(Tp) ≤ p− 2:

p− 1p− 1

p− 2p− 2p− 2p− 2p− 2

A = n vertices

Then|∂A||A| =

(n− 2)(p− 2) + 2(p− 1)

n−−−→n→∞

p− 2.




From Cheeger constant to expanders

Properties of the discrete Cheeger constant:

◮ Two graphs G,G′ with V = V′ and E ⊂ E′. Then h(G) ≤ h(G′) (moreedges/higher connectivity yields larger Cheeger constant...)







◮ α = h(G) can be interpreted as edge expansion rate: given any set A of kvertices, then there are at least αk edges connecting them with vertices inthe complement.








We are looking for families of increasing finite graphs Gn with uniform lowerbound on their Cheeger constants:

h(Gn) ≥ C > 0.

But,...








We are looking for families of increasing finite graphs Gn with uniform lowerbound on their Cheeger constants:

h(Gn) ≥ C > 0.

But,...at the same time the number of edges should not increase too fast(linearly with the number of vertices).




Definition of expander graphs

DefinitionA sequence Gn = (Vn,En) of connected finite graphs with |Vn| → ∞ is called afamily of expanders if






◮ all Gn are p-regular graphs (⇒ |En| =p2 |Vn|)






◮ all Gn are p-regular graphs (⇒ |En| =p2 |Vn|)

◮ h(Gn) ≥ C > 0 for all n




Existence and explicit construction of expanders

◮ Existence of expander families: Using counting arguments, it can be shownfor k ≥ 5 that, for large enough n, most k-regular graphs with n vertices

have Cheeger constant ≥ 12 (Pinsker 1973).







◮ Explicit construction of expanders is difficult.








First explicit construction was given by Margulis 1973 (based on Kazhdan’sproperty (T)):









The graphs Gm = (Vm,Em) are 8-regular graphs with vertex set

Vm = Zm × Zm.










Vm = Zm × Zm.

Let

T1 =

(1 20 1

)

, T2 =

(1 02 1

)

, e1 =

(10

)

, e2 =

(01

)

.










Vm = Zm × Zm.

Let

T1 =

(1 20 1

)

, T2 =

(1 02 1

)

, e1 =

(10

)

, e2 =

(01

)

.

Then every v ∈ Vm is connected to T1v, T2v, T1v+ e1, T2v+ e2.




Margulis’ example (1973)

The graph G4 = (Z4 × Z4,E4). Recall v ; T1v,T2v,T1v+ e1,T2v+ e2.

(0,0) (0,1) (0,2) (0,3)

(1,0)

(1,1)

(1,2)

(1,3)

(2,0)

(2,1)

(2,2)

(2,3)

(3,0)(3,1) (3,2)

(3,3)




Part 2: Spectral properties of expanders

Adjacency matrix of a finite graph G = (V,E):






v1 v2

v3 v4

Label the vertices as v1, . . . , vn.






v1 v2

v3 v4

Label the vertices as v1, . . . , vn. Define symmetric n× n matrix AG = (aij) by

aij =

{

1, if vi ∼ vj,

0, otherwise.






v1 v2

v3 v4

Label the vertices as v1, . . . , vn. Define symmetric n× n matrix AG = (aij) by

aij =

{

1, if vi ∼ vj,

0, otherwise.

AG =

0 1 1 11 0 0 11 0 0 11 1 1 0




Spectrum and spectral gap

Spectrum of a graph G:

σ(G) = {eigenvalues of AG},= {λ1,λ2, . . . ,λn}

with λ1 ≥ λ2 ≥ · · · ≥ λn (counting with multiplicities).








Facts:

◮ If G is p-regular then λ1 = p and λ1, . . . ,λn ∈ [−p, p]. In this case:








Facts:


◮ G is connected iff λ2 < p.








Facts:



◮ G is bipartite iff λn = −p.








Facts:



◮ G is bipartite iff λn = −p.

DefinitionThe spectral gap of a connected, p-regular finite graph G is given by p− λ2(G).

−p p = λ1λ2 < p0




Spectral description of expanders

Theorem (Dodziuk ’84, Alon-Milman ’85, Alon ’86)

G = (V,E) finite, connected p-regular graph. Then

p− λ2

2≤ h(G) ≤

√

2p(p− λ2).







p− λ2

2≤ h(G) ≤

√

2p(p− λ2).

; Alternative definition of expander family: Sequence of increasingp-regular graphs Gn with p− λ2(Gn) ≥ C > 0.







p− λ2

2≤ h(G) ≤

√

2p(p− λ2).

; Alternative definition of expander family: Sequence of increasingp-regular graphs Gn with p− λ2(Gn) ≥ C > 0.

Gabber-Galil showed 1981 for Margulis’ example that

λ2(Gm) ≤ 5√2 < 8,

so spectral gaps of all Margulis graphs are at least 8− 5√2.




Part 3: New examples of expander families

◮ Lubotzky-Phillips-Sarnak, Margulis: Ramanujan Graphs






◮ Reingold-Vadham-Wigderson, Rozenman-Shalev-Wigderson :expanders via zig-zag product







◮ Kassabov : Symmetric groups and expanders








◮ Kassabov- Lubotzky-Nikolov and Breuillard-Green-Tao: Finite simplegroups as expanders









◮ Bourgain-Gamburd : expanders and SL2(p)









◮ Bourgain-Gamburd : expanders and SL2(p)

◮ Lubotzky-Samuels-Vishne , Sarveniazi : expanders and Euclideanbuildings

In this talk: new families of expanders(joint work with N.Peyerimhoff and with R.Blok and C.Hoffman)




Theorem 1 (N.P., A.V.)

The groupsΓn = 〈x, y | r1, r2, r3, [y, x, . . . , x

︸︷︷︸

n

]〉

with

r1 = yxyxyxy−3x−3,

r2 = yx−1y−1x−3y2x−1yxy,

r3 = y3x−1yxyx2y2xyx

are finite, satisfy |Γn| → ∞, and the associated Cayley graphs Cay(Γn, {x, y}) are anexpander family of 4-regular graphs.

Nice properties: the underlying groups have only two generators and fourrelations.





There is a family of finite nilpotent groups Nk, generated by two generators xk.yk,such that |Nk| = 2nk with strictly increasing nk → ∞. The Cayley graphsGk = Cay(Nk, {xk, yk}) are a family of 4-regular expanders forming a tower ofcoverings

· · · → G3 → G2 → G1 → G0.

v0

v1

0

w1

w2

w3

w12

w13

w23

w123

v0*v1

v0*v1+w1

v0*v1+w3

v0*v1+w2

v0*v1+w12

v0*v1+w13

v0*v1+w23

v0*v1+w123

v0+w1

v1+w1

v0+w2

v1+w2

v0+w3

v1+w3

v0+w12

v1+w12

v0+w13

v1+w13

v0+w23

v1+w23

v0+w123

v1+w123 (The graph G2)





The pro-2 completion of the group

Γ0 = 〈x, y | r1, r2, r3〉

satisfies the Golod-Shafarevich inequality

|R| ≥ |S|24

,

is infinite, not 2-adic analytic, contains a free subgroup of rank two but not a freepro-2 subgroup.




The construction

We start with the infinite group

Γ = 〈x0, x1, . . . , x6 | x0x1x3, x1x2x4, . . . , x6x0x2〉.

Let S = {x0, x1, . . . x6}. Then Cay(Γ,S) is a one-skeleton of a thick Euclideanbuilding with the following properties:

◮ lots of (equilateral) triangles (chambers)




The construction


Γ = 〈x0, x1, . . . , x6 | x0x1x3, x1x2x4, . . . , x6x0x2〉.



◮ every vertex has degree 14




The construction


Γ = 〈x0, x1, . . . , x6 | x0x1x3, x1x2x4, . . . , x6x0x2〉.




◮ 3 triangles meet at every edge




The construction


Γ = 〈x0, x1, . . . , x6 | x0x1x3, x1x2x4, . . . , x6x0x2〉.





◮ any two triangles ∆1,∆2 lie in a common plane (apartment) tessellatedby equilateral triangles




The construction


Γ = 〈x0, x1, . . . , x6 | x0x1x3, x1x2x4, . . . , x6x0x2〉.






◮ link of every vertex is the incidence graph of a finite projective planewith 7 points




The construction


Γ = 〈x0, x1, . . . , x6 | x0x1x3, x1x2x4, . . . , x6x0x2〉.






◮ link of every vertex is the incidence graph of a finite projective planewith 7 points

Γ belongs to a family considered by Edjvet,Howie 1989, and with relation tobuildings by Cartwright/Steger/Mantero/Zappa 1993.




On SQ-universality of small cancellation groups

The star graph of P is the graph Γ with vertex set the disjoint union X ∪X−1,and an edge from x to y for each distinct word x−1yw that is a cyclicpermutation of an element of R∪ R−1. The edges corresponding to x−1ywand y−1xw−1 form an inverse pair. Note that a relator of the form x3, say,gives rise to exactly one inverse pair of edges between x and x−1, and thateach inverse pair of edges contributes a single directed edge in Γ.





Definition 1. A presentation P = 〈X | R〉 is defined to be (m, k)-special forpositive integers m > 2 and k > 3 if it is a finite presentation such that thefollowing conditions hold:

(i) the star graph Γ of P is a connected, bipartite graph of diameter m andgirth 2m in which each vertex has degree at least 3;







(ii) each relator r ∈ R has length k; and







(ii) each relator r ∈ R has length k; and

(iii) if m = 2 then k > 4.





The group G is defined to have a special presentation if G can be defined by an(m, k)-special presentation for some m and k. (We remark that our definitionis inspired by Jim Howie whose definition of special presentation coincideswith being (3, 3)-special.)





Recall that a countable group G is SQ-universal if every countable group canbe embedded in a quotient of G.

Question (Jim Howie, 1989) Are groups with (3, 3)-special presentation areSQ-universal?

The answer is negative (Martin Edjvet, AV)




Link and Kazdhan property (T)

The link of the vertex e ∈ Γ:

x0

x1

x2

x3

x4

x5

x6

x−10

x−11

x−12

x−13

x−14

x−15

x−16

Interpreting xi as POINTS and x−1i as LINES yields the incidence relations of

a finite projective plane.




Link and Kazdhan property (T)

The link of the vertex e ∈ Γ:

x0

x1

x2

x3

x4

x5

x6

x−10

x−11

x−12

x−13

x−14

x−15

x−16

Interpreting xi as POINTS and x−1i as LINES yields the incidence relations of

a finite projective plane.

Zuk 1996, Ballmann/Swiatkowski 1997 ⇒ Γ has Kazhdan property (T)

(Cartwright/Młotkowski/Steger 1993: Γ has Kazhdan property (T))




Kazhdan property (T)

DefinitionA locally compact group Γ has Kazhdan property (T) if any unitary representation ofΓ which has almost invariant vectors has an invariant unit vector.

Explanation: A unitary representation ρ : Γ → U(H) (with H a Hilbert spaceover C) has almost invariant vectors if for any compact K ⊂ Γ and ǫ > 0 thereis a unit vector v ∈ H such that ‖v− ρ(g)v‖ < ǫ for all g ∈ K.

Monograph:Bekka, de la Harpe, Valette: Kazhdan’s Property (T), Cambridge 2008




The constructionThe subgroup of Γ, generated by x0, x1, is an index two subgroup givenabstractly by

Γ0 = 〈x0, x1 | r1, r2, r3〉,where

r1 = x1x0x1x0x1x0x−31 x−3

0 ,

r2 = x1x−10 x−1

1 x−30 x21x

−10 x1x0x1,

r3 = x31x−10 x1x0x1x

20x

21x0x1x0.





Γ0 = 〈x0, x1 | r1, r2, r3〉,where

r1 = x1x0x1x0x1x0x−31 x−3

0 ,

r2 = x1x−10 x−1

1 x−30 x21x

−10 x1x0x1,

r3 = x31x−10 x1x0x1x

20x

21x0x1x0.

We also use an explicit representation of the group Γ0 by

x0 = A0 +A11

t, x1 = B0 + B1

1

t,

where A0,A1 are certain 9× 9 matrices over F2.





Γ0 = 〈x0, x1 | r1, r2, r3〉,where

r1 = x1x0x1x0x1x0x−31 x−3

0 ,

r2 = x1x−10 x−1

1 x−30 x21x

−10 x1x0x1,

r3 = x31x−10 x1x0x1x

20x

21x0x1x0.

We also use an explicit representation of the group Γ0 by

x0 = A0 +A11

t, x1 = B0 + B1

1

t,

where A0,A1 are certain 9× 9 matrices over F2.

Rewriting this representation with finite band Toeplitz matrices andestablishing certain periodicity patterns for higher commutators are at theheart of the proofs of Theorems 1 and 2.




We have x0 = A0 + 1yA1 and x1 = B0 + 1

yB1 with 9× 9 matrices

A0,A1,B0,B1 ∈ M(9,F2), and their inverses x−10 , x−1

1 are of the same form.Therefore, an arbitrary group element x ∈ G is of the form

x = C0 +k

∑j=1

1

yjCj,

which we identify with the (finite band) upper triangular infinite Toeplitzmatrix




x =

C0 C1 C2 . . . Ck 0 0 . . .

0 C0 C1 . . . Ck−1 Ck 0. . .

0 0 C0 . . . Ck−2 Ck−1 Ck

. . ....

. . .. . .

. . .. . .

. . .. . .

. . .

, (1)

where each Ci is a matrix in M(9,F2). One checks that multiplication ofelements in GL(9,F2[1/y]) and of the corresponding infinite matrices isconsistent.




A conjecture

MAGMA calculations support the following

Conjecture (N.P., A.V.)

Let λn(Γ0) be the lower exponent-2 central series of Γ0, i.e.,

λn+1 = [λn, Γ0]λ2n.

Then we have the following 3-periodicity for the abelian quotients λn/λn+1, n ≥ 2:

|λn/λn+1| =

{

8, if n ≡ 0, 1 mod 3,

4, if n ≡ 2 mod 3.




A conjecture




λn+1 = [λn, Γ0]λ2n.


|λn/λn+1| =

{

8, if n ≡ 0, 1 mod 3,

4, if n ≡ 2 mod 3.

This conjecture would imply that the tower of coverings of expander graphsin Theorem 2 would always have very low covering indices4, 8, 4, 4, 8, 4, 4, 8, 4, 4, 8, 4, 4, . . . .




A conjecture




λn+1 = [λn, Γ0]λ2n.


|λn/λn+1| =

{

8, if n ≡ 0, 1 mod 3,

4, if n ≡ 2 mod 3.




New girth and finite simple groups

DefinitionLet X = (V,E) be a finite k-regular graph with n vertices. we say that X is an(n, k, c) expander if for any subset A ⊂ V

|∂A| ≥ c(1− |A|N

)|A|

where ∂A = {v ∈ V | d(v,A) = 1}.





DefinitionLet X = (V,E) be a finite k-regular graph with n vertices. we say that X is an(n, k, c) expander if for any subset A ⊂ V

|∂A| ≥ c(1− |A|N

)|A|

where ∂A = {v ∈ V | d(v,A) = 1}.

Theorem (Margulis)

Let Γ be a finitely generated group that has property (T). Let L be a family of finiteindex normal subgroups of Γ and let S = S−1 be a finite symmetric set of generatorsfor γ. Then the family {X(Γ/N,S) | N ∈ L} of Cayley graphs of the finite quotientsof Γ with respect to the image of S is a family of (n, k, c) expanders forn = |Γ/N|, k = |S| and some fixed c > 0.





DefinitionLet Γ,L,S as above. Consider also the natural map φN : X(Γ,S) → X(Γ/N,S).The new girth of a graph X(Γ/N,S) is the length of the shortest circuit γ inX(Γ/N,S) so that γ is not the image of a circuit in X(Γ,S) under the map φN .





DefinitionLet Γ,L,S as above. Consider also the natural map φN : X(Γ,S) → X(Γ/N,S).The new girth of a graph X(Γ/N,S) is the length of the shortest circuit γ inX(Γ/N,S) so that γ is not the image of a circuit in X(Γ,S) under the map φN .

Theorem (R.Blok,C.Hoffman,AV)

For any n there exists an ǫ > 0 and a symmetric set Sn,q of generators for SU2n(q)so that Sn,q has size five and the family of Cayley graphs X(SU2n(q),Sn,q) for q ≥ nforms an ǫ-expanding family of unbounded new girth.




Beauville surfaces

DefinitionA Beauville surface S is a complex algebraic surface, of the form(C1 × C2)/G, where C1 and C2 are non-singular projective curves of generag(Ci) > 2, and G is a finite group acting freely on the product of curves byholomorphic transformations.




Beauville surfaces

DefinitionA Beauville surface S is a complex algebraic surface, of the form(C1 × C2)/G, where C1 and C2 are non-singular projective curves of generag(Ci) > 2, and G is a finite group acting freely on the product of curves byholomorphic transformations.

DefinitionA surface S is said to be isogenous to a product is S admits finite unramifiedcovering isomorphic to a product of curves.




Beauville surfaces

DefinitionLet S be a surface isogenous to a product with minimal realisationS ∼= (C1 × C2)/G. We say that S is a mixed case if the action of G exchangesthe two factors (and then C1 and C2 are isomorphic) and an unmixed case if Gacts via a diagonal action.




Beauville surfaces

Beauville’s original example had two curves C1 = C2, given by the Fermatcurve x5 + y5 + z5 = 0, and G the group (Z/5Z)2 acting on C1 × C2 by therule

(a, b) · ([x : y : z], [u : v : w]) = ([ξax : ξby : z], [ξa+3bu : ξ2a+4bv : w]),

where ξ = e2πi5 and a, b ∈ Z/5Z. Then S is a Beauville surface of unmixed

type with g(C1) = g(C2) = 6.




Beauville surfaces

Let G be a finite group and r an integer with r > 2. An r-tuple T = [g1, ..., gr]of elements of G is called a spherical system of generators, if g1, ..., gr generate Gand we additionally have g1...gr = 1.For an r-tuple T = [g1, ..., gr] of elements of G and g ∈ G, we set

gTg−1 := [gg1g−1, ..., ggrg

−1].

If A = [m1, ...,mr] is an r-tuple of natural numbers with 2 6 m1 6 ... 6 mr,then the spherical system of generators T = [g1, ..., gr] is said to be of type A, ifthere is a permutation τ ∈ Sym(r) such that we have

ord(g1) = mτ(1), ord(g2) = mτ(2), · · · , ord(gr) = mτ(r).

(Here ord(g) is the order of the element g ∈ G.)




Beauville surfaces

For a spherical system of generators T = [g1, ..., gr] of G, we define

Σ(T) = Σ([g1, ..., gr]) :=⋃

g∈G

∞⋃

j=0

r⋃

i=1

{g · gji · g−1} (2)

to be the union of all conjugates of the cyclic subgroups generated by theelements g1, ..., gr. A pair of spherical systems of generators (T1,T2) of G iscalled disjoint if

Σ(T1) ∩ Σ(T2) = {1}.




Beauville surfaces

Next, we introduce unmixed and mixed ramification structures.

DefinitionLet A1 = [m(1,1), ..,m(1,r)] and A2 = [m(2,1), ..,m(2,s)] be tuples of naturalnumbers with 2 6 m(1,1) 6 ... 6 m(1,r) and 2 6 m(2,1) 6 ... 6 m(2,s). An

unmixed ramification structure of type (A1,A2) for G is a disjoint pair (T1,T2) ofspherical systems of generators, such that T1 has type A1 and T2 has type A2.

The disjointness of the pair (T1,T2) in the definition of an unmixedramification structure guarantees that G acts freely on the product CT1

× CT2

of the associated algebraic curves




Beauville surfaces

DefinitionLet A = [m1, ...,mr] be an r-tuple of natural numbers with 2 6 m1 6 ... 6 mr.A mixed ramification structure of type A for G is a pair (H,T) where H is asubgroup of index 2 in G and T = [g1, ...gr] is an r-tuple of elements of G suchthat the following hold:




Beauville surfaces


◮ T is a spherical system of generators of H of type A,




Beauville surfaces



◮ for every g ∈ G \H, the spherical systems T and gTg−1 are disjoint,




Beauville surfaces



◮ for every g ∈ G \H, the spherical systems T and gTg−1 are disjoint,

◮ for every g ∈ G \H we have g2 6∈ Σ(T).




Beauville surfaces

DefinitionAn unmixed Beauville structure is an unmixed ramification structure with twospherical systems (T1,T2) of generators of length 3, i.e., r = 3 and s = 3.A mixed Beauville structure is a mixed ramification structure with a sphericalsystem T of generators of length 3, i.e., r = 3.




Beauville surfaces

Let G1, G2,... be the lower 2-central series for the group Γ0 andH1,H2,... be the lower 2-central series for Γ.

Theorem (N.Barker,N.Boston,N.Peyerimhoff,AV)

Let 3 ≤ k ≤ 64. If k is not a power of 2, then Γ/Hk admits an unmixed Beauvillestructure.




Beauville surfaces

Let G1, G2,... be the lower 2-central series for the group Γ0 andH1,H2,... be the lower 2-central series for Γ.


Let 3 ≤ k ≤ 64. If k is not a power of 2, then Γ/Hk admits an unmixed Beauvillestructure.


Let 3 ≤ k ≤ 10. If k is not a power of 2, then Γ0/Gk admits a mixed Beauvillestructure.



a. vdovina newcastle university oxfordpeople.maths.ox.ac.uk/drutu/conference/vdovinaslides.pdf ·...

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