aa r X i v : . [ m a t h . C O ] F e b Non-expander Cayley graphs of simple groups
G´abor SomlaiAlfr´ed R´enyi Institute of Mathematicsemail: [email protected] ∗ Abstract
For every infinite sequence of simple groups of Lie type of growingrank we exhibit connected Cayley graphs of degree at most 10 such thatthe isoperimetric number of these graphs converges to 0. This proves thatthese graphs do not form a family of expanders.
Let G be a finite group and T a subset of G . The Cayley graph Cay ( G, T ) isdefined by having vertex set G and g is adjacent to h if and only if g − h ∈ T .A Cayley graph Cay ( G, T ) is undirected if and only if T = T − , where T − = (cid:8) t − ∈ G | t ∈ T (cid:9) .Let Γ be an arbitrary graph and S ⊆ V (Γ). We define the boundary of S which we denote by ∂S to be the set of vertices in V (Γ) \ S with at least oneneighbour in S . For a graph Γ the isoperimetric number h (Γ) is defined by h (Γ) = min (cid:26) | ∂S || S | (cid:12)(cid:12)(cid:12)(cid:12) S ⊂ V (Γ), 0 < | S | ≤ | V (Γ) | (cid:27) .A graph Γ is called an ǫ -expander if h (Γ) ≥ ǫ and a series of k -regular graphsΓ n is called an expander family if there is a constant ǫ > n the graph Γ n is an ǫ -expander. Finally, we say that a family of groups G n isa family of uniformly expanding groups if there exist 0 < ǫ ∈ R and k ∈ N suchthat for every i and every generating set S i ⊂ G i of size at most k the Cayleygraphs Cay ( G i , S i ) are ǫ -expanders.The study of series of Cayley graphs of finite simple groups has receivedgreat attention. The proof of the fact was completed by Kassabov, Lubotzkyand Nikolov in [3] based on several earlier work (see [4],[5],[6],[8],[12]), thatthere exist k ∈ N and 0 < ǫ ∈ R such that every non-abelian finite simple groupwhich is not a Suzuki group has a set of generators S of size at most k for which Cay ( G, S ) is an ǫ -expander. This work was extended by Breuillard, Green and ∗ Research is partially supported by MTA Renyi ”Lendulet” Groups and Graphs ResearchGroup G n is a sequence ofnon-abelian simple groups such that the rank of G n is unbounded, then for every n there exists a generating set T n ⊂ G n such that the graphs Cay ( G n , T n ) donot form a family of expanders. An explicit example (see [9]) of a non-expanderfamily of Cayley graphs of special linear groups was given by Luz. The diameterof the graphs given by Luz was investigated by Kassabov an Riley and it wasproved in [7] that there exists c ∈ R such that the diameter of the graphs issmaller than c log ( | SL ( n, p ) | ). Similarly, the symmetric group S n is generatedby γ = (12) and σ n = (1 , , . . . , n ) for every n ∈ N and the sequence of isoperi-metric numbers h ( Cay ( S n , γ, σ n )) tends to 0, see [9]. Moreover, one can finda set of generators of S n such that the diameter of the corresponding Cayleygraphs is Ω( n ) which gives that these Cayley graphs do not form a family ofexpanders.We will investigate 7 series ( A l , B l , C l , D l , A n − , A n , D n ) of finite simplegroups of Lie type. These are the groups of Lie type such that the rank of asequence of groups tends to infinity if we fix a series of the Lie group. In orderto define generators and subgroups of these groups we will use the generatorsgiven by Steinberg in [13] and we will use the notation and several results of thebook of Carter [2].For these 7 series of finite simple groups of Lie type we construct Cayleygraphs and subsets such that the number of the neighbours of these subsetsdepends on the rank of the groups. Moreover, the isoperimetric number ofthese graphs tends to 0. This proves the conjecture of Lubotzky concerning theseries of Cayley graphs of simple groups of unbounded rank. More precisely, weprove the following: Theorem 1. (a) Let G be a Chevalley group of rank l of type A l , B l , C l or D l . For every l ≥ and for every finite field GF ( q ) there exists agenerating set T of cardinality at most and a subset of the vertices S ⊂ V ( Cay ( G, T )) with | S | ≤ | G | such that | ∂ ( S ) || S | ≤ l − .(b) Let G be a twisted group of type A n − , D n or A n . For every n ≥ andfor every finite field GF ( q ) there exists a generating set T ′ of cardinalityat most and S ′ ⊂ V ( Cay ( G, T ′ )) with | S ′ | ≤ | G | such that | ∂ ( S ′ ) | | S ′ | ≤ n − . The paper is organized into the following 4 sections. In Section 2 we give allnecessary definitions and we collect some important fact about the constructionof simple groups of Lie type. The proof of Theorem 1 (a) is contained in Section3 and Theorem 1 (b) which is the case of twisted groups will be handled inSection 4. In Section 5 we present the original construction in terms of matriceswhich was extended to several different series of simple groups.2
Preliminaries
In this section we collect important facts about about finite simple groups ofLie-type and we build up the notation we will use all along this paper.Let K = GF ( q ) be a finite field. We denote by Φ the system of roots andΦ = Φ + ∪ Φ − is the union of the positive and negative roots. We also chooseΠ = { r , r , . . . , r l } ⊂ Φ + which is the set of the fundamental roots.The Weyl group W is generated by the fundamental reflections w r , where r ∈ Π. In order to simplify notation we denote by w i the fundamental reflections w r i , where r i ∈ Π. We denote by x r ( ψ ) the standard generators of the Chevalleygroup G , where r ∈ Φ and ψ ∈ K . If r = r i for some r i ∈ Π, then we denoteby x i ( ψ ) the standard generator x r i ( ψ ). These elements generate the Chevalleygroup G . The subgroups X r = { x r ( t ) | t ∈ K } are called root subgroups of G if r ∈ Φ.The Weyl group W is isomorphic to N/H for some H ⊳ N ≤ G . The cosetsof H in N can be written as n w H for all w ∈ W and N is generated by H and the elements n r for r ∈ Φ. Moreover, n r = x r (1) x − r ( − x r (1) is theelement of the subgroup generated by the root subgroups X r and X − r . It iswell known that n r x s ( t ) n − r = x w r ( s ) ( η r,s t ) for some η r,s ∈ K depending onlyon r and s , see [2, p.101.]. The elements of the normal subgroup H of N canbe written in the form h ( χ ) where χ is a K -character of Z Φ. The subgroup H is generated by the elements of the set { h r ( λ ) | r ∈ Φ and λ ∈ K ∗ } , wherethe K -character corresponding to h r ( λ ) is χ r,λ with χ r,λ ( a ) = λ a,r )( r,r ) . H is anormal subgroup of N and n w h ( χ ) n − w = h ( χ ′ ), where χ ′ ( r ) = χ ( w − ( r )) see[2, p.102.]. Furthermore, h r ( λ ) = n r ( λ ) n r ( −
1) and hence h r ( λ ) ∈ h X r , X − r i ,see [2, p.96.]. In this section we construct series of Cayley graphs for 4 different series ofChevalley groups. For these Chevalley groups we need 6 series of Cayley graphs.The six different constructions are similar but we will treat them separately.We first prove the following technical lemma.
Lemma 1.
Let w = w w . . . w l be a Coxeter element of the Weyl group W andlet us assume that the fundamental root r i is orthogonal to r j if i + 1 < j ≤ l and r i +1 is orthogonal to r k if ≤ k ≤ i − . We also assume that r i and r i +1 have the same length and w i ( r i +1 ) = r i + r i +1 . Then w ( r i ) = r i +1 .Proof. Since r i is orthogonal to r j for every j > i + 1 we have that w ( r i ) = w w . . . w i w i +1 ( r i ). The elements w k are reflections through the hyperplaneperpendicular to r k . Thus w k ( r k ) = − r k for every 1 ≤ k ≤ l and w i +1 ( r i ) = r i + r i +1 = w i ( r i +1 ) since r i and r i +1 have the same length. It follows that w i w i +1 ( r i ) = w i ( r i + r i +1 ) = w i ( r i ) + w r i +1 = − r i + ( r i + r i +1 ) = r i +1 . Hence w ( r i ) = w w . . . w i − ( r i +1 ) = r i +1 since r i +1 is orthogonal to r k for 1 ≤ k ≤ i − (cid:4) .1 A l Let G be a Chevalley group of type A l . The Dynkin diagram of the correspond-ing root system is the following. ❜ r ❜ r ❜ r . . . ❜ r l − ❜ r l One can see from the Dynkin diagram that w i ( r i +1 ) = r i + r i +1 = w i +1 ( r i )for i = 1 , . . . , l − w = w w . . . w l be a Coxeter element of the Weyl group. We choose λ to be a generator of the multiplicative group of GF ( q ). Lemma 2. x (1) , n w and h r ( λ ) generate the Chevalley group G .Proof. It was proved in [13] that x (1) n w and h r ( λ ) generate G . Clearly, x (1)and n w generate x (1) n w which proves the Lemma. (cid:4) For every l ≥ a = Cay ( G, (cid:8) x (1) , n w , h r ( λ ) , x (1) − , n − w , h r ( λ ) − (cid:9) ).Let K a be the subgroup of the Chevalley group G generated by the rootsubgroups X r , X − r , X , X − r , . . . , X r l − , X − r l − and let S a = ∪ l − i =0 K a n iw .Every element of the Weyl group W acts on the the root system Φ. Lemma 3.
The orbit of w which contains r is the following: ✲ r ✲ w r ✲ w r ✲ w . . . ✲ w r l − ✲ w r l ✲ w − r − . . . − r l ✲ wThis can be formulated as follows: w ( r i ) = r i +1 f or ≤ i ≤ l − w ( r l ) = − r − . . . − r l w ( − r − r − . . . − r l ) = r Proof.
Lemma 1 gives that w ( r i ) = r i +1 for 1 ≤ i ≤ n − w ( r l ) = w w . . . w l ( r l ) = w w . . . w l − ( − r l ) = − w w . . . w l − ( r l )since w is a linear transformation of the vector space spanned by the roots. Wealso have w j ( r j +1 + · · · + r l ) = r j + r j +1 + · · · + r l for 1 ≤ j ≤ l −
1. Therefore w w . . . w l − ( r l ) = w w . . . w l − ( r l − + r l )= w w . . . w l − ( r l − + r l − + r l ) = · · · = r + r + . . . + r l .This shows that w ( r l ) = − ( r + r + . . . + r l ). (1)4sing again the linearity of w and equation (1) we get w ( r + r + . . . + r l ) = r + r + . . . + r l − ( r + r + . . . + r l ) = − r .This gives w ( − ( r + r + . . . + r l )) = r , finishing the proof Lemma 3. (cid:4) It follows from Lemma 3 that if 1 ≤ i ≤ l −
1, then n iw K a n − iw contains n iw X r l − i n − iw = X r l . Therefore n iw K a n − iw = K a which shows that n iw / ∈ K a forevery 1 ≤ i ≤ l −
1. This implies that K a , K a n w , . . . , K a n l − w are different rightcosets of K a so S a is the union of l pairwise disjoint subsets of the vertices ofΓ a and these subsets have the same cardinality. Lemma 4. | ∂ ( S a ) || S a | ≤ l Proof. S a is the union of l right cosets of K a so | S a | = l | K a | . It is clear fromthe definition of S a that ( K a n iw ) n w ⊂ S a for every 0 ≤ i ≤ l − (cid:0) K a n iw (cid:1) n − w ⊂ S a if 1 ≤ i ≤ l −
1. Therefore those neighbors of S a which arenot in S a can only be obtained as an element of following subset of the verticesof Γ a : K a n lw [ K a n − w l − [ i =1 (cid:0) K a n iw (cid:1) x (1) l − [ i =1 (cid:0) K a n iw (cid:1) x (1) − l − [ i =1 (cid:0) K a n iw (cid:1) h r ( λ ) l − [ i =1 (cid:0) K a w i (cid:1) h r ( λ ) − . K a is a subgroup of G so (cid:0) K a n iw (cid:1) x = K a n iw if and only if n iw xn − iw ∈ K a .We first apply this observation to x (1) and x (1) − = x ( − n iw x ( ± n − iw is of the form x w i ( r ) ( α ) = x i +1 ( α ) forsome α ∈ GF ( q ) ∗ if 0 ≤ i ≤ l −
1. It follows that n iw x ( ± n − iw ∈ X r i +1 ⊂ K a if i = l − h r ( λ ) and h r ( λ ) − are in the subgroup h X r , X − r i we getthat n iw h r ( λ ) ± n − iw ∈ h X w i ( r ) , X − w i ( r ) i = h X r i +1 , X − r i +1 i ⊂ K a if i = l − ∂S a ⊆ K a n lw ∪ K a n − w ∪ K a n l − w x (1) ∪ K a n l − w x ( − ∪ K a n l − w h r ( λ ) ∪ K a n l − w h r (cid:0) λ (cid:1) . These subsets are all of them right cosets of K a so they havethe same cardinality which proves that | ∂S a | ≤ | K a | , while | S a | = l | K a | . (cid:4) Remark 1.
In order to prove Theorem 1 (a) we repeat the previous constructionseveral times. In every single case the connection set of the Cayley graph willconsist of few standard generators of the Chevalley group, an element of theform n w , where w = w w . . . w l is a Coxeter element of the correspondingWeyl group and an element of the group H . If G is of rank l we will choosea subgroup of G which is isomorphic to a Chevalley group of rank l − andwhich is of the same type. The subset of the vertices for which the isoperimetricnumber is sufficiently small will be the union of cosets of the subgroup of rank l − . .2 B l Let G be a Chevalley group of type B l The Dynkin diagram of a Chevalleygroup of type B l is the following: ❡ r ❡ r ❡ r . . . ❡ r l − ❡ r l It is easy to see from the Dynkin diagram that w ( r ) = r + 2 r and w ( r ) = r + r .One can see using Lemma 1 that w ( r i ) = w w . . . w l ( r i ) = r i +1 for 2 ≤ i ≤ l − . (2)The fundamental roots r , . . . , r l are orthogonal to r . Therefore w ( r ) = w w ( r ) = w ( r + r ) = − r + ( r + 2 r ) = r + r . We also have that w is linear so using equation (2) we have that if 2 ≤ j ≤ l −
1, then w ( r + r + . . . + r j ) = w ( r )+ w ( r )+ . . . + w ( r j ) = r + r + r + . . . + r j +1 . (3)Using these observations we conclude that the following picture represents apart of the orbit of the action of the group generated by w including the root r : r ✲ w r + r ✲ w r + r + r ✲ w . . . ✲ r + r + . . . + r l ✲ wThis can be formulated as follows: w i ( r ) = r + r + . . . + r i +1 for i = 1 , . . . , l −
1. (4)The orbit of h w i containing these elements contains w ( r + r + . . . + r l ) aswell. It is easy to see that w i ( r i +1 + r i +2 + . . . + r l ) = r i + r i +1 + . . . + r l if2 ≤ i ≤ l −
1. We also have w ( r ) = 2 r + r hence w ( r l ) = w . . . w l − w l ( r l ) = − w . . . w l − ( r l )= − w . . . w l − ( r l − + r l ) = · · · = − w ( r + . . . + r l )= − ( r l + . . . + r + 2 r ).This implies using equation (3) that w ( r + . . . + r l ) = w ( r + . . . + r l − ) + w ( r l ) = − r . (5)One can easily describe the remaining elements of the orbit since w is linear.We also investigate the action of h w i on 2 r + r + . . . + r l and r + . . . + r l .Using equation (5) and the linearity of w we get that w (2 r + r + . . . + r l ) = w ( r ) + w ( r + r + . . . + r l ) = r + r − r = r . It follows using equation (2)that w i (2 r + r + . . . + r l ) = r i +1 for 1 ≤ i ≤ l −
1. (6)One can also prove using equation (4) and equation (6) that w i ( r + . . . + r l ) = − r + r + . . . + r i +1 ) + r i +1 for 1 ≤ i ≤ l −
1. (7)6 .2.1
Char ( K ) > char ( K ) > Lemma 5. x (1) , n w and h t ( λ ) , where t = 2 r + r + · · · + r l generate theChevalley group G of type B l if the characteristic of the underlying field is not .Proof. It was proved in [13] that x (1) n w and h t ( λ ) generate the Chevalleygroup G if char ( K ) = 2. (cid:4) We define again a sequence of connected Cayley graphs. LetΓ b = Cay (cid:0) G, (cid:8) x (1) , x ( − , n w , n − w , h t ( λ ) , h t ( λ ) − (cid:9)(cid:1) ,where G is of rank l and w = w w . . . w l . Similarly to the previous case let K b = h X r , X − r , X r , X − r , . . . , X r l − , X − r l − i and let S b = ∪ l − i =0 K b n iw . Lemma 6. | ∂ ( S b ) || S b | ≤ l − Proof.
We claim that S b is the union of pairwise disjoint right cosets of thesame subgroup K b in G . We only have to show that n iw / ∈ K b if 1 ≤ i ≤ l − n iw X r l − i n − iw = X r l if1 ≤ i ≤ l −
2. Therefore X r l ⊂ n iw K b n − iw = K b if 1 ≤ i ≤ l − n iw / ∈ K b . Thus S b is the union of l − K b .Using the definition of the Cayley graph Γ b we have that ∂S b is a subset ofthe following set: l − [ i =0 ( K b n iw ) n w l − [ i =0 ( K b n iw ) n − w l − [ i =0 ( K b n iw ) x (1) l − [ i =0 ( K b n iw ) x ( − l − [ i =0 ( K b n iw ) h t ( λ ) l − [ i =0 ( K b n iw ) h t ( λ ) − .By the definition of S b the subsets K b n iw n w are contained in S b if 0 ≤ i ≤ l − K b n iw n − w ⊂ S b if 1 ≤ i ≤ l − n iw x ( ± n − iw = x r + r + ... + r i +1 ( t ) for some t ∈ K ∗ . If 0 ≤ i ≤ l −
2, then x r + r + ... + r i +1 ( t ) ∈ K b since r + r + . . . + r i +1 is in the root system generated by the fundamental roots r , r , . . . , r l − and K b is the Chevalley group of type B l − generated by the corresponding rootsubgroups. Therefore K b n iw x ( ±
1) = K b n iw ⊂ S b if 0 ≤ i ≤ l − h t ( λ ) and h t ( λ ) − = h t ( λ ) are in the subgroup generated by X t and X − t . Equation (6) shows that n iw X t n − iw = X w i ( t ) = X r i +1 and bythe linearity of w we have n iw X − r n − iw = X − r i +1 for i = 1 , , . . . , l −
2. Thus n iw h t ( λ ) n − iw and n iw h t ( λ ) n − iw are in h X r i +1 , X − r i +1 i ≤ K b if 1 ≤ i ≤ l − ∂S b ⊂ K b n l − w ∪ K b n − w ∪ K b h t ( λ ) ∪ K b h t ( λ ) which gives | ∂S b || S b | ≤ | K b | ( l − | K b | = l − . (cid:4) .2.2 Char ( K ) = 2 Lemma 7. x s (1) , x − r (1) , n w and h t ( λ ) , where s = r + · · · + r l generate theChevalley group G of type B l if char ( K ) = 2 .Proof. It was proved in [13] that x s (1) x − r (1) n w and h t ( λ ) generate G if K = GF (2 k ) with k > x s (1) x − r (1) and n w generate G if | K | = 2. (cid:4) Let Γ ′ b = Cay (cid:0) G, (cid:8) x s (1) , x − r (1) , n ± w , h t ( λ ) ± (cid:9)(cid:1) .The set S b can be considered as a subset of V (Γ ′ b ) so we claim the following. Lemma 8. | ∂ ( S b ) || S b | ≤ l − Proof.
It was proved in Lemma 6 that | S b | = ( l − | K b | .Similarly, the proof of Lemma 6 shows that K b n iw h t ( λ ) ± ⊂ S b if 1 ≤ i ≤ l −
2. By the definition of S b we have K b n iw n w ⊂ S b if 0 ≤ i ≤ l − K b n iw n − w ⊂ S b if 1 ≤ i ≤ l − w ( − r ) = − w ( r ) and equation (4) we get that n iw x − r (1) n − iw ∈ K b if 0 ≤ i ≤ l − w i ( r ) = r + r + . . . + r i +1 by equation (4). Hence K b n iw x − r (1) = K b n iw ⊂ S b .Equation (7) shows that n iw x s (1) n − iw is in K b if 1 ≤ i ≤ l −
2. Therefore (cid:0) ∪ l − i =1 K b n iw (cid:1) x s (1) ⊂ S b . Finally, we conclude that ∂ ( S b ) ⊂ K b n − w ∪ K b n l − w ∪ K b h t ( λ ) ∪ K b h t ( λ ) − ∪ K b x s (1). (cid:4) C l The Dynkin diagram is the following in this case: ❡ r ❡ r ❡ r . . . ❡ r l − ❡ r l It can easily be verified using the Dynkin diagram that w l − ( r l ) = r l + 2 r l − and w l ( r l − ) = r l − + r l .Using Lemma 1 one can see that w ( r i ) = r i +1 for i = 1 , , . . . , l −
2. We alsohave w ( r l − ) = w w . . . w l ( r l − ) = w w . . . w l − ( r l + r l − )= w w . . . w l − ( r l + r l − ).Since r l is orthogonal to the remaining roots r , r , . . . , r l − we have w ( r l − ) = r l + w w . . . w l − ( r l − ).Since w i ( r i +1 + . . . + r l − ) = r i + r i +1 + . . . + r l − for i = 1 . . . l − w w . . . w l − ( r l − ) = w w . . . w l − ( r l − + r l − ) = r + . . . + r l − + r l − .This gives w ( r l − ) = r + r + . . . + r l .8sing all these observations we can determine a part of the orbit of h w i containing r , which is the following: ✲ r ✲ w r ✲ w . . . ✲ w r l − ✲ w r l + r l − + r l − + . . . + r ✲ Lemma 9. x (1) , n w and h r ( λ ) generate the Chevalley group G .Proof. The proof can be found in [13]. (cid:4)
The construction is almost the same as in the case A l . LetΓ c = Cay (cid:0) G, (cid:8) x (1) , x ( − , n w , n − w , h r ( λ ) , h r ( λ ) − (cid:9)(cid:1) .Let K c = h X r , X − r , X r , X − r , . . . , X r l , X − r l i and let S c = ∪ l − i =0 K c n iw . Lemma 10. | ∂ ( S c ) || S c | ≤ l − Proof.
Similarly to the previous cases n − iw K c n iw contains n − iw X r i +1 n iw = X r for1 ≤ i ≤ l − n iw is not in K c if 1 ≤ i ≤ l −
2. This proves that | S c | = ( l − | K c | .Again, K c n iw n w ⊂ S c if 1 ≤ i ≤ l − K c n iw n − w ⊂ S c if i = 0.It is also easy to verify that n iw x (1) ± n − iw = ( x i +1 ( t )) ± for some t ∈ GF ( q ) ∗ . Therefore n iw x (1) ± n − iw ∈ X r i +1 and n iw h r ( λ ) ± n − iw are in the sub-group generated by X r i +1 and X − r i +1 for i = 1 , . . . , l −
2. Thus the elementsof the right cosets K c n iw x (1) ± and K c n iw h r ( λ ) ± are in S c if 1 ≤ i ≤ l − ∂S c ⊆ K c n l − w ∪ K c n − w ∪ K c x (1) ∪ K c x (1) − ∪ K c h r ( λ ) ∪ K c h r ( λ ) − , which is the union of 6 right cosets of K c . Thus | ∂S c | ≤ | K c | . (cid:4) D l The Dynkin diagram in this case is the following: ❡ r ❅❅ ❡ r (cid:0)(cid:0) ❡ r ❡ r . . . ❡ r l − ❡ r l Lemma 11. (a) x r (1) , n w and h r ( λ ) generate the Chevalley group G if therank of G is odd.(b) x − r (1) , x r (1) , x (1) , n w and h r ( λ ) generate the Chevalley group G ifthe rank of G is even.Proof. The proof can be found in [13]. (cid:4) h w i which contains r . The root r is orthogonal to r , . . . , r l hence w ( r ) = w w w ( r ) so we have w w w ( r ) = w w ( r + r ) = w ( r + r + r ) = − r + r + r + r = r + r and similarly w ( r ) = w w w ( r ) = w w ( r + r ) = w ( − r + r + r ) = r + r .Using Lemma 1 we get w ( r i ) = r i +1 for i = 3 , . . . , l −
1. This gives that both w i ( r ) and w i ( r ) are of the form r i +2 + r i +1 + . . . + r + y , (8)where y = r or y = r . Let us assume that l is odd.Let Γ d = Cay (cid:0) G, (cid:8) x r (1) , x r (1) − , n w , n − w , h r ( λ ) , h r ( λ ) − (cid:9)(cid:1) .Let K d = h X r , X − r , X r , X − r , . . . , X r l − , X − r l − i and let S d = ∪ l − i =0 K d n iw . Lemma 12. | ∂ ( S d ) || S d | ≤ l − Proof.
It is easy to see that if 0 ≤ i ≤ l −
3, then n iw x r (1) ± n − iw = x w i ( r ) ( t ) ± ∈ K d for some t ∈ GF ( q ) ∗ since by (8) the root w i ( r ) is a linear combinationwith integer coefficients of the fundamental roots r , r , . . . , r l − and similarly n iw h r ( λ ) ± n − iw ∈ K d . It follows that ∂S d ⊆ K d n l − w ∪ K d n − w .It remains to show that for S d is the union of l − K d . Again, n iw K d n − iw contains the subgroup n iw X r l − i n − iw = X r l if 1 ≤ i ≤ l − n iw / ∈ K d . (cid:4) Let us assume that l is even.LetΓ ′ d = Cay (cid:0) G, (cid:8) x − r ( ± , x − r ( ± , x r ( ± , n w , n − w , h r ( λ ) , h r ( λ ) − (cid:9)(cid:1) .Let K ′ d = h X r , X − r , X r , X − r , . . . , X r l − , X − r l − i and let S ′ d = ∪ l − i =0 K ′ d n iw .10 emma 13. | ∂ ( S ′ d ) || S ′ d | ≤ l − Proof.
It is clear that w i ( − r ) = − w i ( r ) and hence w i ( − r ) is in the rootsystem generated by the roots r , r , . . . , r l − if 1 ≤ i ≤ l −
4. This shows that n iw x ± r ( ± n − iw = x w i ( ± r ) ( t ) ∈ K ′ d for some t ∈ GF ( q ) ∗ .It was proved in Lemma 12 that n iw h r ( λ ) ± n − iw ∈ K ′ d if 0 ≤ i ≤ l − n iw x r ( ± n − iw = x r i ( t ) for some t ∈ K ∗ which is in K ′ d if 0 ≤ i ≤ l −
4. It follows that ∂S ′ d ⊆ K ′ d n l − w ∪ K ′ d n − w .It remains to show that S ′ d is the union of l − K ′ d . This is clear since if 1 ≤ i ≤ l −
4, then n iw K ′ d n − iw contains the subgroup X r l which shows that n iw / ∈ K ′ d . (cid:4) The twisted groups can be obtained as subgroups of Chevalley groups. In orderto define twisted groups we need to find a non-trivial symmetry ρ of the Dynkindiagram. We restrict our attention to those twisted groups which are definedusing a symmetry of order 2 and we also assume that the roots in Φ have thesame length. It is well known that such an symmetry ρ can be extended to aunique isometry τ of V which is the vector space spanned by Φ. We assume that Aut ( GF ( q ) ∗ ) contains an element of order 2. Then the Chevalley group G hasan automorphism of order 2, which we denote by α such that x r ( t ) α = x r ( k )for every r ∈ ± Π and k ∈ K , where k = τ ( k ) and r = ρ ( r ).The subgroup U is the set of elements u ∈ U such that u α = u and similarly V = { v ∈ V | v α = v } . The twisted group G is generated by U and V . Thesubgroups H and N are defined as the intersection of G with H and N ,respectively. We denote by W the elements w of the Weyl group W such that τ wτ − = w . There is a natural isomorphism of the group W to N /H andwe denote by n w the element of N ≤ N which corresponds to w ∈ W .The set of positive roots Φ + has a partition where the elements of the par-tition are of the following form: Z = (cid:8) r | r ∈ Φ + (cid:9) if r = rZ = (cid:8) r, r | r ∈ Φ + and r + r / ∈ Φ (cid:9) Z = (cid:8) r, r, r + r | r ∈ Φ + and r + r ∈ Φ (cid:9) .We denote by Π the collection of sets which are elements of the partition. Foreach set Z in the partition there is a unique element w Z ∈ W which is generatedby { w r | r ∈ Z } such that w ( Z ) = − Z . These elements are the following: w Z = w r if Z = (cid:8) r | r ∈ Φ + (cid:9) w Z = w r w r if Z = (cid:8) r, r | r ∈ Φ + and r + r / ∈ Φ (cid:9) w Z = w r + r = w r w r w r if Z = (cid:8) r, r, r + r | r ∈ Φ + and r + r ∈ Φ (cid:9) .11very element of Π can be obtained as w ( Z ), where w ∈ W and Z containsa fundamental root. Those sets which contain a fundamental root are calledfundamental sets. Moreover, W is generated by (cid:8) w Z | Z ∈ Π (cid:9) .For every Z ∈ Π we denote by X Z the subgroup generated by the rootsubgroups X r for r ∈ Z and X Z = X Z ∩ G . A n − The fundamental sets in this case are the following: Z n = { r n } , Z i = { r i , r n − i } for 1 ≤ i ≤ n − W are: w Z n = w n , and w Z i = w i w n − i for 1 ≤ i ≤ n − X Z = (cid:8) x r ( t ) | t = t (cid:9) if Z = { r } X Z = (cid:8) x r ( t ) x r ( t ) | t ∈ K (cid:9) if Z = { r, r } .Let n w = n w n w . . . and h e = h r ( t ) h r ( t ), where t generates K ∗ . In thefollowing in order to simplify notation we write n w instead of n w .We also define x e = x r (1) x r n − (1) which is an element of X Z and which canalso be written as x r (1) x r (1) α = x r (1) x r (1). Lemma 14. x e , n w and h e generate the group G .Proof. The proof can be found in [13]. (cid:4)
Let Γ e = Cay (cid:0) G, (cid:8) x e , x − e , n w , n − w , h e , h − e (cid:9)(cid:1) .Let K e = h X Z , X − Z , X Z , X − Z , . . . , X Z n , X − Z n i and let S e = ∪ n − i =0 K e n iw . K e can be considered as a twisted group which is a subgroup of the Cheval-ley group generated by the root subgroups X r , X − r , . . . , X r n − , X − r n − . Thecorresponding set of fundamental roots is ρ -invariant and we denote by Φ n − the root system generated by these roots. The restriction of ρ to the set { r , r , . . . , r n − } gives a symmetry of the Dynkin diagram of these roots whichextends to an isometry. This isometry is the restriction of τ . This gives thatfor Z ∈ Π the subgroup X Z is a subgroup of K e if and only if Z ⊂ Φ n − .Clearly, h r ( t ) is in h X Z , X − Z i ⊂ G if Z = { r } with r = r and if Z = { r, r } ,then there is homomorphism of SL ( K ) onto h X Z , X − Z i ⊂ G which showsthat x r ( t ) x r ( t ) ∈ G and h r ( t ) h r ( t ) ∈ G .12onjugating by n iw ∈ N we get the following: n − iw X Z n iw = n − iw ( X Z ∩ G ) n iw = n − iw X Z n iw ∩ n − iw G n iw = X w − i ( Z ) ∩ G = X w − i ( Z ) . (9) Lemma 15. | ∂ ( S e ) || S e | ≤ n − Proof.
We claim that S e is the union of n − K e n jw = K e n j ′ w if and only if n j − j ′ w ∈ K e so we have to show that n iw / ∈ K e if 1 ≤ i ≤ n − w i ( r ) = r i +1 if 1 ≤ i ≤ n −
2. If k ≤ n −
3, then w ( r k ) = w w n − . . . w k w n − k w k +1 ( r k )since r k is orthogonal to r j if j ≥ k + 2. Therefore w ( r k ) = w w n − . . . w k ( r k + r k +1 )= w w n − . . . w k − ( r k +1 ) = r k +1 since r k +1 is orthogonal to the roots r n − k , . . . , r n − and r k +1 is orthogonalto r , . . . , r k − . It follows that w − i ( Z i +1 ) = Z and hence by equation (9) X Z ⊂ n w − i S e n iw if 1 ≤ i ≤ n −
2. This proves that n iw / ∈ K e if 1 ≤ i ≤ n − | S e | = ( n − | K e | .It is easy to see that S e contains K e n iw n w if i = 0 , , . . . , n − S e contains K e n iw n − w if i = 1 , , . . . , n − K e n iw g = K e n iw if and only if n iw gn − iw ∈ K e .Since n iw x r (1) n − iw = x w ( r ) ( λ ) for some λ ∈ K and x e = x r (1) x r (1) α we have n iw x r (1) x r (1) α n − iw = n iw x r (1) n − iw n iw x r (1) α n − iw = n iw x r (1) n − iw ( n iw x r (1) n − iw ) α = x w i ( r ) x w i ( r ) ( λ ) α = x r i +1 x r i +1 ( λ ) α .This shows that n iw x e n − iw ∈ X Z i +1 which proves that if i = 1 , , . . . , n −
2, then n iw x ± e n − iw ∈ K e and hence K e n iw x ± e = K e n iw since Z i +1 ⊂ Φ n − .We also have n iw h r ( t ) h r ( t ) n − iw = h r i +1 ( θ ) h w i ( r ) ( θ ′ ) for some θ , θ ′ ∈ K .Using the fact that w ∈ W we have w i ( r ) = w i ( r ) so h r i +1 ( θ ) h w i ( r ) ( θ ′ ) = h r i +1 ( θ ) h r i +1 ( θ ′ ). Clearly, n iw h e n − iw ∈ H . Thus θ ′ = θ and n iw h ± e n − iw = (cid:16) h r i +1 ( θ ) h ( r i +1 ) ( θ ) (cid:17) ± ∈ K e since r i +1 ∈ Φ n − if i = 1 , . . . , n −
2. This provesthat K e n iw h ± e = K e n iw if i = 1 , . . . , n − ∂S e ⊂ K e n n − w ∪ K e n − w ∪ K e x e ∪ K e x − e ∪ K e h e ∪ K e h − e , finishing the proof of Lemma 15. (cid:4) D n The fundamental sets in this case are the following: Z = { r , r } , Z i = { r i +1 } for 2 ≤ i ≤ n − W are: w Z = w w , and w Z i = w i +1 for 2 ≤ i ≤ n − n w = n w n w . . . n w n − and h f = h r ( t ) h r ( t ), where t generates K ∗ .We also define x f = x r (1) x r (1) which can also written as x r (1) x r (1) α = x r (1) x r (1). Lemma 16. x f , n w and h f generate the group G .Proof. The proof can be found in [13]. (cid:4)
Let Γ f = Cay (cid:16) G, n x f , x − f , n w , n − w , h f , h − f o(cid:17) .Let K f = h X Z , X − Z , X Z , X − Z , . . . , X Z n − , X − Z n − i and let S f = ∪ n − i =0 K f n iw .We denote by Φ n − the root system generated by the fundamental roots r , r , . . . , r n − . Lemma 17. | ∂ ( S f ) || S f | ≤ n − Proof.
The Coxeter element in this case is exactly the same as in subsection3.4. This gives that n iw ( r n − i ) = r n for 0 ≤ i ≤ n −
3. The fundamen-tal sets Z , Z , . . . , Z n − consist of only one element thus n iw S f n − iw contains X w i ( Z n − − i ) = X w i ( r n − i ) = X r n = X Z n − if 1 ≤ i ≤ n − S f contains X n − i . This proves that if 1 ≤ i ≤ n −
3, then n iw / ∈ K f . Thus S f is the unionof n − | S f | = ( n − | K f | .Using the definiton of S f one can see that K f n iw n w ⊂ S f if i = 0 , , . . . , n − K f n iw n − w ⊂ S f if i = 1 , . . . , n − n iw x f n − iw are of the form x r ( t ) x r ( ± t ) for some r ∈ Φ and t ∈ K ∗ . In order to prove that these elements are in K f for i = 0 , , . . . , n − r ∈ Φ n − . Using the fact that the Coxeter element inthis case is the same as in Section 3.4 we have that both w i ( r ) and w i ( r ) areof the form r + r + r + . . . + r i +1 or r + r + r + . . . + r i +1 . These roots areclearly in the root system generated by the fundamental roots r , r , . . . , r l − if i ≤ n −
2. This proves that n iw x ± f n − iw is in K f if 0 ≤ i ≤ n − S f x ± f ⊂ S f .Similarly, the elements n iw h f n − iw are of the form h r ( t ) h r ( t ) for some r ∈ Φand t ∈ K ∗ and it is easy to see that r ∈ Φ n − if 0 ≤ i ≤ n −
3. This provesthat n iw h ± f n − iw is in K f if 0 ≤ i ≤ n − S f h ± f ⊂ S f . (cid:4) .3 A n The fundamental sets are the following: Z = { r n , r n +1 , r n + r n +1 } , Z i = { r n +1 − i , r n + i } for 2 ≤ i ≤ n .Let n w = n w n w . . . n w n and h g = h r n ( t ) h r n ( t ), where t generates K ∗ .We also define x g = x r n (1) x r n +1 (1) x r n + r n +1 ( k ) with k + k = 1. Lemma 18. x g , n w and h g generate the group G .Proof. The proof can be found in [13]. (cid:4)
Let Γ g = Cay (cid:0) G, (cid:8) x g , x − g , n w , n − w , h g , h − g (cid:9)(cid:1) .Let K g = h X Z , X − Z , X Z , X − Z , . . . , X Z n − , X − Z n − i and let S g = ∪ n − i =0 K g n iw . Lemma 19. | ∂ ( S g ) || S g | ≤ n − Proof.
First, we show that S g is the union of n − n iw / ∈ K g for i = 1 , . . . , n −
2. This willbe done by proving that X Z n is contained in n iw K g n − iw . Using equation (9) weonly have to show that w i ( Z n − i ) = Z n for i = 1 , . . . n − r k +1 is contained in Z n − k . Let us assume that 1 ≤ k ≤ n − w ( r k +1 ) = w n w n +1 w n w n − w n +2 . . . w w n ( r k +1 )= w n w n +1 w n w n − w n +2 . . . w k +1 w n − k w k ( r k +1 )since r k +1 is orthogonal to the roots r j if j > n or j < k −
1. Clearly, w k +1 w n − k w k ( r k +1 ) = r k so w ( r k +1 ) = w n w n +1 w n . . . w k +2 w n − k − ( r k ) = r k since the remaining reflections fix r k .One can see by induction that w i ( r i +1 ) = r for i = 1 , . . . n − w ∈ W we have w i ( r i +1 ) = w i ( r i +1 ) = r n and hence w i ( Z n − i ) = Z n . This provesthat for i = 1 , . . . , n − n iw ( K g ) n − iw contains X Z n . Therefore | S g | = ( n − | K g | .The definition of S g shows that K g n iw n w ⊂ S g if i = n − K g n iw n − w ⊂ S g if i = 0. It remains to investigate the elements of the form n iw x ± g n − iw and n iw h ± g n − iw . 15e claim, that w i ( r n ) = r n + r n − + . . . + r n − i if i ≤ n −
2. Using theorthogonality of the fundamental vectors r j , r k , where | j − k | ≥ w ( r n ) = w n w n +1 w n w n − w n +2 . . . w w n ( r n )= w n w n +1 w n w n − ( r n ) = w n w n +1 ( r n − ) = r n − + r n . (10)Similarly, if 1 ≤ k ≤ n −
2, then w ( r n − k ) = w n w n +1 w n w n − w n +2 . . . w w n ( r n − k )= w n w n +1 w n . . . w n − k w n + k +1 w n − k − ( r n − k )= w n w n +1 w n . . . w n − k +1 ( r n − k − ) = r n − k − . (11)Since w is linear we get using (10) and (11) that w i ( r n ) = r n + r n − + . . . + r n − i . (12)By observing equations (12) one can see that if i = 0 , . . . , n −
2, then both r and r n are orthogonal to w i ( r n ) and similarly r and r n are orthogo-nal to w i ( r n +1 ) = w i ( r n ) = w i ( r n ) = r n + i +1 . This shows that for w ′ = w n w n +1 w n w n − w n +2 . . . w w n − we have w i ( r n ) = ( w ′ ) i ( r n ) and w i ( r n +1 ) =( w ′ ) i ( r n +1 ). Therefore w i ( r n + r n +1 ) = ( w ′ ) i ( r n + r n +1 ). Moreover, n iw x g n − iw = n iw ′ x g n − iw ′ and n iw h g n − iw = n iw ′ h g n − iw ′ .Clearly, n w ′ ∈ K g and hence the elements n iw ′ x ± g n − iw ′ and n iw ′ h ± g n − iw ′ are in K g if i = 0 , , . . . n − (cid:4) In order to finish the proof of Theorem (1) we have to verify that the forthose sets S for which boundary ∂ ( S ) is relatively small we have | S | ≤ | G | . Theorder of the investigated simple groups is the following: A l ( q ) : n +1 ,q − q n ( n − Q ni =1 (cid:0) q i +1 − (cid:1) B l ( q ) : ,q − q n Q ni =1 (cid:0) q i − (cid:1) C l ( q ) : ,q − q n Q ni =1 (cid:0) q i − (cid:1) D l ( q ) : ,q n − q n ( n − Q ni =1 (cid:0) q i − (cid:1) A l ( q ) : n +1 ,q +1) q n ( n − Q ni =1 (cid:16) q i +1 − ( − i +1 (cid:17) D l ( q ) : ,q n +1) q n ( n − ( q n + 1) Q ni =1 (cid:0) q i − (cid:1) It is easy to see that such a simple group can not have a subgroup of index atmost 2 l , finishing the proof of Theorem 1. In this section we give explicit generators of the Cayley graphs that we investi-gated in Section 3 and 4. We also show how to find the subsets of the vertices S for which ∂S is relatively small. We only handle the case of Special Linear16roups which can easily be transformed to the case of the Projective SpecialLinear Groups which is clearly the easiest one. This example includes the orig-inal idea which was extended to several different series of simple groups. Inorder to show the simplicity of the original construction we forget about themachinery which was built up before.Let A l = ,where A l ∈ GF ( q ) ( l +1) × ( l +1) . Let B l = . . . ( − l . . . .We denote by C l the diagonal matrix diag ( λ , λ, , , . . . , ∈ GF ( q ) ( l +1) × ( l +1) ,where λ generates GF ( q ) ∗ .We denote by e i,j the matrix with 1 in the ( i, j )-th position and zeros ev-erywhere else and let T i,j ( δ ) = I + δe i,j , where I denotes the identity matrix.Using this notation we can write A l = T , (1).The standard generator x r (1) of the Chevalley group given in Subsection3.1 corresponds to the matrix A l and the Coxeter element n w can be identifiedwith B l . Finally, C l plays the role of h r ( λ ).Clearly, T i,j ( α ) T i,j ( β ) = T i,j ( α + β ) and [ T i,j ( α ) , T j,k ( β )] = T i,k ( αβ ) if i = k ,where [ g, h ] = g − h − gh denotes the commutator of g and h . Lemma 20.
For every l ∈ N the set { A l , B l , C l } forms a generating set of SL ( l + 1 , q ) .Proof. We fix the size of the matrices and hence we can write A = A l , B = B l and C = C l . Let H = h A, B, C i . It is enough to verify that T i,j ( δ ) ∈ H forevery i = j and δ ∈ GF ( q ).It is easy to see that A C k = T , (1) C k = T , ( λ k ). Using T , ( µ ) T , ( η ) = T , ( µη ) we get that T , ( δ ) ∈ h A, C i ≤ H for every δ ∈ GF ( q ). For i = j wehave B k T i,j ( δ ) B − k = T i + k,j + k ( ± δ ), where the indices are taken modulo l + 1and hence T i,i +1 ( δ ) ∈ H for every 1 ≤ i ≤ l and for every δ ∈ GF ( q ). Thisimplies that for every 1 < l ≤ l + 1 and for every δ ∈ GF ( q )[ . . . [[ T , ( δ ) , T , (1)] , T , (1)] . . . , T k − ,k (1)] = T ,k ( δ ) ∈ H .Using again the fact that B k T ,l ( δ ) B − k = T k,l + k ( ± δ ) we get that T i,j ( δ ) ∈ H for every i = j and for every δ ∈ GF ( q ). (cid:4) S = ((cid:18) D E (cid:19) ∈ SL ( l + 1 , q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D ∈ SL ( l, q ) , E ∈ SL (1 , q ) ) .For every 1 ≤ i ≤ l we define S i = S B i .Finally, let S = l − [ i =0 S i .It is easy to see that | S | < | SL ( l +1 ,q ) | if l ≥ Lemma 21. | ∂ ( S ) || S | ≤ l Proof.
Every element of S has exactly l columns with 0 in the last row, exactly1 column with 0 in the first l and 1 in the last row. The sets S i are pairwisedisjoint since an invertible matrix can not have a column with only zero en-tries. Furthermore, they all have the same cardinality since S is a subgroup of SL ( n, q ) and S i are right cosets of S in SL ( n, q ).It is easy to see that SB \ S ⊆ S B l = S l and SB − \ S ⊆ S B − . Theremaining elements of ∂S are of the form M A , M C and
M A − , M C − where M ∈ S .Let us assume that M ∈ S i . Then M = (cid:18) D D ′ (cid:19) for some D ∈ GF ( q ) l,l − i and D ′ ∈ GF ( q ) l,i . Multiplying a matrix M by A or A − from the right only modifies the second column of M . Therefore if M ∈ S i with i = l, l −
1, then it is easy to see that
M A, M A − ∈ S i .Multiplying a matrix M by C or C − from the right only modifies the firstand the second columns of M thus if M ∈ S i with i = l, l −
1, then
M C ± ∈ S i .This gives that ∂S ⊆ S l ∪ S B − ∪ S l − A ∪ S l − A − ∪ S l − C ∪ S l − C − since S = ∪ l − i =0 S i . (cid:4) Acknowledgement
The author is grateful to L´aszl´o Pyber for many valuable suggestions during theresearch.
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