EXPANSION OF PRODUCT REPLACEMENT GRAPHS ... - CiteSeerX
EXPANSION OF PRODUCT REPLACEMENT GRAPHS Alexander Gamburd
MSRI Berkeley, CA 94720 E-mail: [email protected] Igor Pak
Department of Mathematics MIT, Cambridge, MA 02139 E-mail: [email protected] March 25, 2001 Abstract. We establish a connection between the expansion coecient of the product replacement graph ?k (G) and the minimal expansion coecient of a Cayley graph of G In particular, we show that the product replacement graphs ? with k generators. ?k PSL(2; p) form an expander family, under assumption that all Cayley graphs of PSL(2; p), with at most k generators are expanders. This gives a new explanation of the outstanding performance of the product replacement algorithm and supports the speculation that all product replacement graphs are expanders [LP,P3].
Introduction Expanders are highly connected sparse graphs of great interest in computer science, in areas ranging from parallel computation to complexity theory and cryptography; recently they were also used as a key ingredient in connection with the Baum-Connes conjecture [G2] and in computational group theory [LP]. The explicit constructions of expander graphs (by Margulis [M1, M2] and Lubotzky, Phillips, and Sarnak [LPS] ) use deep tools (Kazhdan's property (T), Selberg's Theorem, proved Ramanujan conjectures) to construct families of Cayley graphs of nite groups. The fundamental problem, raised by Lubotzky and Weiss [LW], is whether being an expander family is a property of the groups alone, independent of the choice of generators (Independence Problem). The product replacement algorithm is a commonly used heuristic to generate random group elements in a nite group. Let G be a nite group generated by at most d elements. The product replacement graph ?k (G) is de ned to be a graph, with vertices corresponding to generating k-tuples in G, and edges corresponding to Nielsen transformations. While ?k (G) is closely related to Cayley graphs of G, Key words and phrases. Random walks, mixing time, expander graphs, Independence Problem.
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ALEXANDER GAMBURD, IGOR PAK
these graphs can be de ned as the Schreier graphs of a special group automorphisms of a free group Aut+ (Fk ). Most recently, graphs ?k (G) became a subject of an intense investigation, prompted by the study of a commonly used `practical' algorithm for generating random elements in nite groups, designed Leedham-Green and Soicher [LG]. This algorithm, based on the random walk on product replacement graphs, showed a remarkable performance, as reported in [C et al.] It was suggested in [LP], and, in fact, proved in several special cases, that the product replacement graphs ?k (G) are expanders, for a xed k, when jGj ! 1. The main result of this paper is a theorem, establishing the connection between the expansion coecient of the product replacement graph ?k (G) and the minimal expansion coecient of a Cayley graph of G with k generators. In particular, we show that if one assumes that all Cayley graphs with at most four generators in PSL(2; p) have a universal lower bound on expansion, then the product replacement graphs ?k (PSL(2; p)) form an expander family, when k 8 is xed, and p ! 1. Let ? be a k-regular (oriented) graph with an adjacency matrix A. For the rest of the paper we assume that ? is symmetric, i.e. that A = AT . Consider a nearest neighbor random walk W = W (?), with transition matrix P = A=k. Denote by 1 = 0 > 1 2 : : : the eigenvalues of P, and let (?) = 1 ? 1 be the eigenvalue gap of the graph ?. We say that a sequence of k-regular graphs f?ng is an expander family, if for some > 0, we have (?n ) > , for all n 1. Among many properties of expanders are the bounds on the isoperimetric constant (see below), diameter of the graph diam(?n ) C1 log j?n j, and the mixing time of the random walk mix W (?n ) C2 log j?n j, for some universal constants C1 ; C2 > 0. Let G be a nite group, and let S be a generating set with k elements. We will always assume that S is symmetric : S = S ?1 . Denote by C = C (G; S ) the corresponding Cayley graph on G. Consider ? a nearest neighbor random walk W (G; S ) = W (C ). Denote by (G; S ) = C the eigenvalue gap of C (G; S ). As in case of general graphs, we say that a family of Cayley graphs fCn = C (Gn ; Sn )g is an expander, if there exist " > 0, such that (Cn ) > " for all n. For a xed integer m, we say that a sequence of groups fGn g has universal expansion with m generators, if there ? exist > 0, such that for every n and every hS i = Gn , jS j m, we have C (Gn ; Sn ) > 0. The positive answer to the Independence Problem for PSL(2; p) is the main assumption in this paper: 1) Does the sequence of groups fPSL(2; p); p ? primeg have universal expansion with m = 4 generators? 1 An armative answer to problem 1 is supported by numerical experiments [LR1,LR2] and some recent results [S1,Ga]; see comments in section 6. Let us de ne the product replacement graph ?k (G) as follows. Let vertices be all generating k-tuples of the group G, and let edges correspond to transformations : Li;j and Ri;j : (g1 ; : : : ; gi ; : : : ; gk ) ! (g1 ; : : : ; gi g1 ; : : : ; gk ); Ri;j j 1 Li;j : (g1 ; : : : ; gi ; : : : ; gk ) ! (g1 ; : : : ; gj gi ; : : : ; gk ): 1 In can be shown, in fact, that if 1) holds, then fPSL(2; p)g has universal expansion for every xed m 4 (cf. section 6.)
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These graphs were introduced in [C et al.], in connection with computing in nite groups. Note that graphs ?k (G) are regular, of degree D = 4 k (k ? 1). Let fGn g be a sequence of nite groups, generated by at most d elements, and let k d be xed. As before, we say ? that a sequence f?k (Gn )g is an expander family, if for some > 0 we have ?k (Gn ) > for all n. The main question of our study is the following open problem: ? 2) Does the sequence of graphs ?k PSL(2; p) ; p ? prime form an expander family , for any xed k 3 ? We prove that a positive answer to question 1) implies a positive answer to question 2), for all k 8. In fact, we prove a general result, for every nite group G. We show that, under certain conditions, the Cheeger constant of ?k (G) is bounded from below by the minimal Cheeger constant of the Cayley graph C (G; S ), with jS j k. This idea is similar in spirit to the paper [DS3], where the eigenvalue gap (?k (G)) was bounded in terms of maximal diameter of C (G; S ) (cf. [P3]). For k = (log jGj), the dependence on diameter was later removed in [P4]. Let us say a few words about the proof. The proof is combinatorial in nature and is almost entirely self contained. At the end, we rely upon some results on the group structure of PSL(2; p), which are known in the literature (see below). We use a novel combinatorial technique based on graph decomposition, as opposed to path arguments used in previously in [CG,DS2,DS3,P4]. It is easy to see that such a technique can never prove that a certain family of graphs is an expander family (cf. [P3]). The rest of the paper is structured as follows. In section 1 we state the main results of the paper. Preliminary observations and lemmas are presented in sections 2, and 3. These follow with the proof of main results (section 4) and proof of the lemmas (section 5). We conclude in section 6 with a collection of historical and mathematical remarks, and pointers to the literature. Throughout the paper, [n] will denote f1; 2; : : :; ng. We use G to denote a nite group, and ? to denote a connected regular graph. 1. Main results
Let G be a nite group, d = d(G) be the minimal number of generators of G. We say that the set of generators S is minimal, if no proper subset of S generates G. By m(G) denote the maximal size of the minimal generating set of G. By `(G) denote the length of the maximal subgroup chain of G. Clearly, d(G) m(G) `(G) log2 jGj: Let 'k (G) denotes the probability that k random group elements generate G. Let (G) be the smallest k such that 'k (G) > 1 ? . It was shown in [P1] that (G) `(G) + C log(1=), for a universal constant C > 0. Let ? be an (oriented, loops are allowed) graph. Denote by deg(?) the maximal in and out-degree of a vertex in ?. We say that ? is k-regular if every vertex has in and out-degree k = deg(?). De ne (edge) expansion e(?) as follows: ) ( E? (X; X ) j ? j e(?) = min k jX j : X ?; 1 jX j 2 ;
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where E? (X; Y ) = f(x; y) 2 ? : x 2 X; Y 2 Y g is the set of edges between X and Y , and k = deg(?). Note that 1 > e(?) > 0. The Cheeger-Buser inequality (in this context, also known as conductance bound of Jerrum and Sinclair [JS]) gives:
e(?) (?) e(?) 8 : 2
Thus, a uniform lower bound on expansion e(?n ) > > 0, for a family of k-regular graphs f?n g, is an equivalent de nition of expanders [Lu]. Let C (G; S ) be an (oriented) Cayley graph on G, with a generating set S . Denote by k (G) the smallest expansion of the Cayley graph on G with at most k generators:
?
k (G) = min e C (G; S ) : hS i = G; jS j k : Let ?k (G) be the product replacement graph, de ned as in the introduction Let D = deg(?k (G)) = 4 k (k ? 1).
Main Theorem Let G be a nite group. For every k 2 m(G), there exist = (k) > 0, such that if k 2 (G), then ? e ?k (G) > c k (G); where c = c(k) is a constant, which depends only on k, and not on G. I Note that the result in the theorem holds for every nite group G, not a family of groups. Recall that for any sequence fGn g of simple groups, with jGn j ! 1, we have '2 (G) ! 1, as n ! 1 (see section 2 below). Therefore, for every such sequence fGn g, and > 0, we have (Gn ) ! 2, as n ! 1. The following corollaries follow from Main Theorem.
Corollary 1. Let fGng be a family of nite simple groups, such that jGn j ! 1, as n ! 1. Suppose also that m(Gn ) m, and m (Gn ) > 0, for all n 1. Let k 2 m, D = 4k(k ? 1). Then a family of D-regular graphs ?k (Gn ) is an expander family. I Corollary 2. Let k 8 be a xed integer, and let D = 4k(k ? 1). Assume that there exists > 0, such that ? 4(PSL(2 ; p)) , for all prime p 5. Then a family of D-regular graphs ?k PSL(2; p) is an expander family. I Corollary 3. Let fGng be a family of nite groups, such that `(Gn ) `, for all n ! 1. Suppose also that ` (Gn ) > 0, for all n 1. There exists a universal constant C > 0, such that for all k 2 ` + C log `, the family of 4k(k ? 1)-regular graphs ?k (Gn ) is an expander family. I The Main Theorem and the corollaries will be proved in section 4.
Remark 1. The product replacement graphs of simple groups, studied in Corollary 1, seem to complement the set of graphs ?k (G) that are known to be
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expanders. Indeed, the only other cases, when ?k (G) are shown to be expanders, are the abelian groups and nilpotent groups of bounded nilpotency class [LP]. But in these cases the Cayley graphs have large diameter and cannot be expanders (see [B+] and section 6 below.) Also, although Corollary 3 is stated in general terms, it can, in fact, be applied to variety of algebraic groups (see section 2 below.) 2. Combinatorics and probability on finite groups
Let G be a nite group, and let
'k (G) = P(hg1 ; : : : ; gk i = G) = j?jkG(jGk )j be the probability that k independent random elements in G generate the whole
group. A major recent result in this direction was completed in a sequence of papers by Dixon [Dx] (see also [B1]), Kantor and Lubotzky [KL], Liebeck and Shalev [LS1,LS2]. Together, these papers prove that '2 (Gn ) ! 1, for any sequence of nite simple groups fGn g, such that jGn j ! 1. While the overall ? result is based on classi cation of nite simple groups, the special cases '2 PSL(2; p) ! 1 as p ! 1, and '2 (An ) ! 1 as n ! 1, are completely elementary. In our notation, # (PSL(2; p) = # (An ) = 2 for all > 0, and p, n large enough. Note, that if #1=2 (G) < r (i.e. 'r (G) < 1=2), we easily have 'k (Gn ) < , for k > C r log(1=), and for some universal constant C > 0 (see [P2]). In this case j?k (G)j > (1 ? )jGjk . It is also known that if ` = `(G), then for all k > ` + C log(1=) we have 'k (Gn ) < , for some universal constant C [P2]. While the bound m(G) `(G) is often sharp, there are examples when m(G) is much smaller than `(G) (see [W1,W2]). While the recent work [W2] calculates for a number of simple groups, we will use only result m(PSL(2; p)) 4. There is little doubt that all our results can be generalized to all series of bounded rank. Note that this condition is crucial, since we trivially have m(PSL(n; p)) n ? 1. Let us note here that `(G) is bounded for a large number of algebraic groups, which extends the Corollary 3 beyond simple groups. Indeed, for a series of algebraic groups fG(p)g of the same rank, over Fp (such as fPSL(n; p)g, when n is xed), the order f (p) = ord(G(p)) is a polynomial in p 3 of a xed degree n2 [Bo]. Thus, the sieve methods in number theory imply that f (p) has at most a bounded number of prime factors for in nitely many primes p (see [HR], chapter 8,9.) Therefore, `(Gp ) < C for in nitely many prime p, where C = C (n) is a xed constant. In particular, for G(p) = PSL(2; p), we have f (p) = ord(G(p)) = 21 p(p ? 1)(p + 1). It is believed [O] that there are in nitely many primes q, such that 6q + 1 and 12q +1 are also primes. Taking p = 12q +1, this gives f (p) = 12p(6p +1)(12p +1), so that `(PSL(2; p)) 6 for in nitely many primes p. On the other hand, one can deduce from [HR] that `(PSL(2; p)) 13 for in nitely many primes p. We will also need the following simple result, which we prove in section 5. Lemma 1. Let 1 > > > 0. Consider a nite group G, and let X G, such that 1 jX j (1 ? )jGj. Then P? g X ? X > jX j > 1 ? 11 ?? ;
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where g is uniform in G. I 3. Edge expansion of graphs
In this section we present some known and some new results on edge expansion of graphs. The lemmas are arranged so that the level of generality roughly increases. Since at no point we need sharp bounds, we do not attempt to optimize the constants. Instead, we present simple proofs of (sometimes, known) lemmas so that their generalization can be obtained with no diculty. Let ? = (V; E ) be a nite (oriented) graph. A graph ?0 = (V 0 ; E 0 ) is called a subgraph of ?, if V 0 V and E 0 E . Let ? be a k-regular graph, and let e(?) be the (edge) expansion of ?, de ned as above. It is often convenient to work with the Cheeger constant of G is de ned to be h(?) = e(?) k.
Lemma 2. Let ? = (V; E ) be a nite k-regular graph, ?1 = (V1 ; E1 ), : : : , ?n = (Vn ; En ) be the subgraphs of ?, such that V = [i Vi , and jVi j > (1 ? )jV j, for some 0 < < 51 and all i 2 [n]. Then h(?) maxf1n; 5g min h(?i ) : i 2 [n] : I This lemma is probably well known, although we were unable to locate the precise reference. We postpone the proof until section 5.
Lemma 3. Let X [M ] [N ], jX j (M N=2). Denote Xi; = X \fig [N ], X;j = X \ [M ] fj g. Then, for some universal constants ; > 0, we have: (i; j ) 2 X : X < (1 ? ) N + (i; j ) 2 X : X < (1 ? ) M i; ;j (~) > jX j: Moreover, for all < we have: jf(i; j ) 2 X : Xi; < (1 ? ) N; i (1 ? ) M gj (~~) + jf(i; j ) 2 X : X;j < (1 ? ) M gj > ( ? ) jX j: I Versions of the rst part of Lemma 3 seem to be known, with roughly the same elementary proof. Since we need the second part as well, we present the proof of lemma for values = 1=10 and = 31=90. While these are probably not optimal, they suce for our purposes. For graphs ?1 = (V1 ; E1 ) and ?2 = (V2 ; E2 ), de ne the cartesian product ? = ?1 ?2 to be the graph ? = (V; E ), such that V = V1 V2 and
?
E = (v1 ; v2 ); (v10 ; v20 ) 2 V 2 : (v1 ; v10 ) 2 E1 ; v2 = v20 or (v2 ; v20 ) 2 E2 ; v1 = v10 :
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Let k1 = deg(?1 ) and k2 = deg(?2 ). Note that deg(?) = k1 + k2 .
Proposition 1. [CT,HT] Let ? = ?1 ?2 be the product of graphs ?1 and ?2 . Let h1 = h(?1 ), h2 = h(?2 ). Then h(?) 21 min h1 ; h2 : I The proof is elementary, and follows from Lemma 3 (perhaps, with a di erent constant.) As we need an extension of the proposition, we include a proof with a constant 1=27 instead of 1=2. Let us also quote, without a proof, a known generalization of this result:
Proposition 2. [CT,HT] Let ? = ?1 ?m be the product of graphs
?1 ; : : : ; ?m . Then
h(?) 21 min h(?i ) : i 2 [m] : I
We say that ?0 = (V 0 ; E 0 ) is a restriction of ? = (V; E ), if V 0 V , and (v1 ; v2 ) 2 E , v1 ; v2 2 V 0 , implies that (v1 ; v2 ) 2 E 0 . ? M ] [N ]; E ) be a k-regular graph. De ne ?i; = Vi; ; Ei; , ?;j = ?V Let; E? =,([with Vi; = fig [N ], V;j = [M ] fj g, to be restrictions of ?. ;j ;j
Lemma 4. There exist constants ; > 0, such that for all 0 < the following holds. Let ? = ([M ] [N ]; E ), k = deg(?). Consider restrictions ?i; and ?;j , de ned as above. De ne
?
h1 = min h ?;j : j 2 [N ] ; ? h2 = min h ?i; : i 2 [(1 ? )M ] : Then h(?) min fh1; h2 g. I
In section 5 we deduce the lemma from our proof of (a weaker version of) Proposition 1. Below we present a nal extension of Lemma 3, tailored to our needs. Let ?i = (Ei ; [N ]); i 2 [M ] be a family of k-regular graphs on [N ]. We say that f?i g has -uniform expansion with Cheeger constant bh, if for all X [N ], such that 1 jX j N=2, we have Ei (X; X ) bh2 jX j, for at least (1 ? ) M di erent i 2 [M ]. Of course, if h(?i ) bh for all i 2 [(1 ? )M ] (cf. Lemma 4), then f?i g has -uniform expansion with Cheeger constant bh.
Lemma 5. There exist constants ; > 0, such that for all 0 < the following holds. Let ? = ([M ] [N ]; E ), k = deg(?). Consider restrictions ?i; and ?;j , de ned as above. De ne
?
h1 = min h ?;j : j 2 [N ] ;
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ALEXANDER GAMBURD, IGOR PAK
and suppose ?i; ; i 2 [M ]g is a family of k-regular graphs, which has -uniform expansion with Cheeger constant bh2 . Then h(?) min fh1; bh2 g. I
Remark 2. Let us note that in Lemmas 3, 4 and 5, we cannot weaken the condition to have -error for both types of restrictions. For example, in Lemma 4, we cannot let j 2 [(1 ?)N ]. Similarly, we cannot allow to have -uniform expansion for a family ?i; as well. Indeed, in these cases there can be very small sets X ? which lie in the intersection of `bad' directions. This is the main reason why we cannot weaken our assumption 1) to a weaker version of it, with all Cayley graphs of PSL(2; p) substituted by random Cayley graphs. 4. Proof of the Main Theorem and Corollaries
Proof of Main Theorem.
Fix 1=2 > > 0, and let n = maxfm(G); (G)g, r = k ? n. Since k 2(G) and k 2 m(G), we obtain r maxf(G); m(G)g. De ne an action of Sk on ?k (G) as follows : (g1 ; : : : ; gk ) = (g(1) ; : : : ; g(k) ), for 2 Sk . Consider a subgraph ?0 with vertices all generating k-tuples (g1 ; : : : ; gk ) 2 ? = ?k (G), such that hg1 ; : : : ; gn i = G, and edges corresponding to transforma , L , such that 1 j n < i k, or 1 i n < j k. Consider also tions Ri;j i;j ? 0 ? , de ned as above, for each coset representative 2 (k; n) = Sk = Sn Sr . Clearly, ?0 ' ?0 for all 2 Sk . We have j?0 j > (1 ? ) j?j, since, by de nition, n (G). Also, for every (g) = (g1 ; : : : ; gk ) 2 ?k (G), there exists 0 2 (k; n), such that (g) 2 0 ?0 (this follows from n m(G)). From Lemma 2, we obtain:
h(?) ?1k min fh(?0 ) : 2 (k; n)g = C h(?0 ); n
for some constant C = C (n; k). Now let us prove that h(?0 ) > cn (G). Think of ?0 as a graph on ?n (G) Gr . For every xed? (g) = (g1 ; : : : ; gn) 2 ?n (G), consider ?0(g); ?0 , the subgraph of ?0 with vertices (g); (h) , where (h) 2 Gr is any r-tuple of elements. De ne ?0;(h) ?0 analogously, for every (h) 2 Gr . We have k0 = deg ?0;(h) = deg ?0(g); = 4 n r. De ne C (G; S ) = C (G; S ) [ C ?1 (G; S ) to be a union of two isomorphic Cayley corresponding to multiplication on the left and on the right: C = (g;graphs g s1 ); (g; s1?g) : g 2 G; s 2 S . Clearly, h(C ) = 2 h(C ). Now,? by de ni tion, ?0(g); = C G; fg1; : : : ; gn g r , and by Proposition 2, we have h ?0(g); 1 h(C ) > k 0 (G). n 2 ? We cannot prove that h ?0;(h) > > 0 for two reasons. First, not all elements (h) 2 Gr are generating sets (although there are > (1 ? )jGjr of them). The graphs ?;(h) are disconnected when (h) 2= ?r (G). Thus, we cannot conclude that Cayley graphs on these r-tuples are expanders. Second, graphs ?0;(h) =
?
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C G; fh1; : : : ; hr g n are not products of (union of) Cayley graphs C , but their
intersection with ?k (G). Thus we cannot use Proposition 2 to bound Cheeger constant. In fact, we cannot do this for any xed (h) 2 Gr . Instead, we use Lemma 5 to establish "-uniform expansion of the family ?0;(h) on ?n (G), for (h) 2 Gr . Indeed, consider rst H = Gn and any subset X H , 1 jX j jH j=2. Now apply Lemma 1 to the group H (taking = 1=2 and = 1=4). We obtain that the di erence in the lemma is > jX j=4, for > jX j=3 di erent g 2 H . Now observe that for uniform (h) 2 Gr , the rst n components (h)0 = (h1 ; : : : ; hn ) in (h) are uniform in H . Multiplication of (g) by (h)0 is a composition of transformations L1;n+1 L2;n+2 : : : Ln;2n . By the symmetry, if the composition has expansion > , at least one of the transformations Li;n+i has expansion > =n (cf. [B2]). Since j?n (G)j > (1 ? )jH j, this gives jg X ? X j \ ?n (G)j > (1=4n ? )jX j = jX j for at least jX j=3 di erent g 2 H . This proves the 1=3-uniform expansion for the family of graphs ?0;(h) , with Cheeger constant > . Now take = minf1=4 1=90; 1=8ng, so as to satisfy the lemmas. From Lemma 5 we conclude that h(?0 ) > C (n; k)n (G). Now the theorem follows from the observations above.
Proof of Corollary 1. Since '2 (Gn ) ! 1, and d(Gn ) = 2, the second condition k 2(Gn ) = 4 is trivial. The corollary now follows immediately from the Main Theorem. Proof of Corollary 2.
This is a special case of Corollary 1. Recall that
m(PSL(2; p)) 4, and the result follows.
Proof of Corollary 3. Recall that m(G) `(G), and '`+t = 1 ? O(1=2t) [P2]. Finally, observe that k (G) (k=`)` (this follows by removing extra edges). In the proof of Main Theorem we need to nd k and , such that '(k=2) 1 ? , and = O(1=k). Since `(Gn ) is bounded, we can solve these two equations by taking t = O(log k). We omit the easy calculation. 5. Proofs of Lemmas
Proof of Lemma 1. Note that E?jg X \ X j =
X
x;x 2X 0
Markov inequality gives
P(gx = x0) = jX j2 jG1 j (1 ? )jX j:
P?jg X ? X j < jX j = P?jg X \ X j > (1 ? )jX j < 11 ?? : This implies the result.
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Proof of Lemma 2. Let X V , 1 jX j jV j=2. Consider subsets Xi = X \ V . Denote Ei (X; Y ) = E? (X; Y ), for X; Y Vi , and let ei = e(?i ). Also, let ki = deg(?i ), k = deg(?). Fix a constant = (1 ? )=2 > 0. Note that 2 < < 1. 5 2 There are two possible cases. Either jX j jV j, or jX j > jV j. We consider them separately. In the rst case, jXi j < 1? jVi j = jVi j=2. Therefore jEi (Xi ; Vi ? Xi )j > ei jXi j ki . Since X [i Xi , there is always i 2 [n], such that jXi j jX j=n. Therefore, for this i we have: i
E? (X; X ) jEi (Xi ; Vi ? Xi )j (ei ki ) jXn j : We conclude:
E? (X; X ) 1 min e k : e(?) = X : 1jmin kn i i i X jjV j=2 k jX j In the second case, we have :
jXi j > ( ? ) jV j ( ? ) 2 jX j > 25 jX j;
jXi j jX j 21 jV j < 2(1 1? ) jVi j < 58 jVi j;
and therefore jVi ? Xi j=jXi j < 1?5=58=8 = 35 . For every i 2 [n], we have :
E? (X; X ) Ei (Xi ; Vi ? Xi ) ei ki min fjXi j; jVi ? Xi jg > 35 ei ki jXi j > 35 ei ki 25 jX j > 51 ei ki jX j: We conclude:
E? (X; X ) 1 min e k : e(?) = X : 1jmin X jjV j=2 k jX j 5k i i i This completes the second case and proves Lemma 2.
Proof of Lemma 3. Let = 1=10 and = 31=90. Denote I = i 2 [M ] : jXi; j < 109 N , J = j 2 [N ] : jX;j j < 109 M . Since jX j MN=2, we have 9 MN 10 N (M ? jI j) < jX j 2 ; which gives jI j > 49 M . Therefore, for every j 2 J , we have 31 jX;j ? I fj gj > 109 M ? 59 M = 31 90 M 90 X;j :
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Now
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X X + X X ? I fj g ;j ;j j 2J j 2= J X > (1 ? ) jX j + 31 X;j = (1 ? ) jX j + 31 jX j 31 jX j;
P = [i2I Xi; + [j2J X;j = j 2= J 90
90
90
where P is equal to the l.h.s. of (~) in the lemma, and
=
P
jX;j j 0: jX j
j 2= J
This proves the rst part (~). The second part follows verbatim, except for a substitution of I by I 0 = I \ [(1 ? )M ], and the constant 31=90 is replaced by (31=90 ? ), as in (~~).
Proof of Proposition 1. Recall that we prove only a weaker version of the proposition, with constant 1=27 instead of 1=2, as in the claim. Suppose ?1 = ([M ]; E1 ), ?2 = ([N ]; E2 ). Let ?i; = fig ?2 , ?;j = ?1 fj g, for all i 2 [M ], j 2 [N ]. Let X ?, 1 jX j j?j=2. As in Lemma 3, let Xi; = ?i; \ X , X;j = ?;j \ X . Consider 9 N ; J = j 2 [N ] : X < 9 M : I = i 2 [M ] : Xi; < 10 ;j 10 Also, let
?
?
Ei; = E1 Xi; ; ?i; ? Xi; ; E;j = E2 X;j ; ?;j ? X;j : By de nition of I and J , for all i 2 I , j 2 J we have:
min Xi; ; ?i; ? Xi; > 19 Xi; ; min X;j ; ?;j ? X;j > 19 X;j : By Lemma 3, we have: E (X; ? ? X ) X E + X E X e k Xi; + X e k X;j i; ;j 2 2 1 1 9 9 i2I j 2J i2I j 2J 0 1 X X minfe1 k1 ; e2 k2 g @ Xi; + X;j A 9
We conclude:
i2I
j 2J
jX j min fe k ; e k g: j X j 19 min fe1 k1 ; e2 k2 g 31 1 1 2 2 90 27
e(?) = X : 1jmin X j MN=2
E (X; ? ? X ) k jX j
271 k min fe1 k1 ; e2 k2 g:
12
ALEXANDER GAMBURD, IGOR PAK
Proof of Lemma 4. Let = 1=27, and = 1=90. Use the second part of Lemma 3. Substitute = 901 to obtain that the r.h.s. of (~~) is at least 3190?1 = 13 . Now note that we never used in the proof of Proposition 1 the fact that ?i; (and, similarly, graphs ?;j ) are isomorphic to each other. Now the proof of the lemma follows verbatim the proof of Proposition 1, with the only di erence that we use (~~) instead of (~), with = 1=90, as above. Proof of Lemma 5. Follows verbatim the proof of Lemma 4. Indeed, notice again that in the proof of Proposition 1 we never used the fact that i is always in the same subset of size (1 ? )M in [M ]. Similarly, for every X [N ] we never used the full expansion of ?i; , but rather Ei; = Ei (X; X ). The rest of the proof remains unchanged. 6. Final Remarks
Let us elaborate on the rich history of the problem and known results, related to both questions 1) and 2) in the introduction. It is well known that, in a certain precise sense, \random" k-regular graphs are expanders. Only a much weaker result is known for Cayley graphs, when k is allowed to grow with jGj. The best known bound for all nite group is the case when k = (log jGj) [AR] (see also [P1]). While this bound cannot be improved for abelian groups, no better result is known for other classes of groups (cf. [B3]). The rst explicit constructions of expanders were found by Margulis [M1], who used Kazhdan's property (T) from representation theory to prove the expansion. The next breakthrough came in papers [LPS,M2], where the authors used harmonic analysis and number theory to obtain the explicit constructions of so called Ramanujan graphs, the expanders with the largest possible eigenvalue gap (when k is xed). Both approaches use Cayley graphs of linear groups, and neither of them is elementary, although an e ort to simplify the technique has been made (see [GG,Lu,DaS].) Most recently, combinatorial constructions of expanders has been introduced in [RVW]. One can think of our results as of new approach to develop expanders. In case of Cayley graphs, only very special generators has been used, although recent improvements increase the variety of such sets (see [Ga,GJS,S1,S2]). These results support an armative answer to the Independence Problem 1) for PSL(2; p), i.e. that all Cayley graphs of PSL(2; p) form an expander family (see [LW]). Further support is given by numerical evidence [LR1,LR2]. Our results indicate the importance of this problem for computational group theory. Let us note Independence Problem remains open even for \random" generating sets [B+,Lu], and there seem to be little hope of proving 1) with existing techniques. On the other hand, it was speculated in [LW] that universal expansion property must hold for all group sequences, which admit some expanding family of Cayley graphs. If true, this would allow us to prove expansion for a large family of product replacement graphs.
EXPANSION OF GRAPHS
13
As we mentioned above, the product replacement graphs ?k (G) in this form were introduced recently in connection with the `practical' algorithm for generating random elements [C et al.]. On the other hand, a related family of graphs ?ek (G) was studied back in the sixties by B.H. Neumann, M. Dunwoody, and others, in connection with the so called T-systems (see [P3] for the references). Many basic questions about these graphs remain unanswered, such as the connectivity of ?k (G), for general nite groups G. In our running example, it was proved by Gilman that graphs ?k (PSL(2; p)) are connected, for k 3, and p 5 [Gi]. In general, it is known that ?k (G) is connected for all k > m(G) + d(G) [DS3,P3]. Now, a rigorous study of convergence of random walks on the product replacement graphs ?k (G), for general nite groups G and in special cases, was undertaken in a number of recent papers [B4,CG,DS2,DS3,LP,P3]. In the latest paper [P4], the second author showed that the random walk mixes in time polynomial in k and log jGj, for k = (log jGj). Still, for small k, the nature of the practical rapid mixing remains unclear. One possible explanation came in [LP], where the authors showed that ?k (G) are always expanders, provided a known open problem 3) has positive solution: 3) Does group Aut(Fk ) have Kazhdan's property (T) ? The problem 3) remains open; an indirect evidence in favor of it is the fact proved in [CV] the it has property (FA) of Serre. It is also known that Aut(Fk ) are hyperbolic and thus nonamenable [G1]. There are also some negative indications: Aut(F2 ) and Aut(F3 ) are shown not to have (T) [Mc], and Aut(Fk ) do not have bounded generation [Su], a property closely related to (T) [S3]. Now, since the authors in [C+] test the product replacement algorithm on a number of simple and quasisimple groups, one can think of this work as an alternative explanation of the algorithm performance. Let us mention here that it was proved (unconditionally) in [LP], that graphs ?k (G) are expanders, when G is nilpotent of class `, and both k and ` are xed. It is entirely possible that any family of graphs ?k (G) , for a xed k, is an expander. While a counterexample to this claim would give a negative answer to 3), a proof of this would not, however, imply 3). We refer to an extensive survey article [P3] for references and details. Let us note that the main theorem is inapplicable to a family of alternating groups fAn g, where n 5. Not all Cayley graphs of An are expanders (see below), and also m(An ) = n ? 2 [W1], which contradicts the assumptions in Corollary 1. Let us present here an important closely related open problem [B+,Lu,LW]: 4) Is there any sequence of Cayley graphs fC (Sn ; Rn ) , which is an expander (for some generating sets hRn i = Sn .) Not unlike question 1), question 4) remains dicult if not unapproachable. Only recently, a sequence of bounded generating sets hRn i = Sn , with diam C (Sn ; Rn ) = O(n log n), has been constructed [BKL,Q]. It was widely speculated that the answer to 4) is negative, i.e. that there are no expanders on Sn [LW]. At the moment, not even generating sets with mix W = O(n log n) are known. The sets Rn , as above, come close with mix W (Sn ; Rn ) = O(n log3 n) [DS1]. To add to a confusion, let us mention here a conjecture that all Cayley graphs on Sn have diameter
14
ALEXANDER GAMBURD, IGOR PAK
at most O(n2 ) [B3,Di], while for \random" Cayley graphs the diameter ?O(ispnbe-) lieved to? be O(n log n ) [K1]. The best bounds in both cases are exp and exp O(log2 n) , respectively [B+,BH]. It is easy to nd a non-expanding family in Sn , i.e. Rn = f(1; 2); (1; 2; : : :; n)1 g, such that diam C (Sn ; Rn ) = (n2 ) (see [Di,DS1,Lu]). Still, for all we know, \random" Cayley graphs on Sn can be expanders [B3,BH]. Let us conclude with an interesting observation in [LP], which connects all questions 1) { 4). First, consider a diagonal action of Aut(G), de ned as follows : (g1 ; : : : ; gk ) = ( (g1 ); : : : ; (gk )), for 2 Aut(G). De ne a graph ?ek (G) with vertices corresponding to orbits of action of Aut(G) and edges corresponding to (see [Gi,P3]). Clearly, if ?k (G) is connected, then ?ek (G) is also Li;j and Ri;j connected [P3]. as of Now, let Fk be a free group on k generators. One can think of Li;j and Ri;j 2 + (special ) Nielsen generators in Aut(Fk ). Let Aut (Fk ) = hLi;j ; Ri;j i Aut(Fk ). It is easy to see that Aut+ (Fk ) is a subgroup of index 2 in Aut(Fk ) [LP,P3]. One can think of graphs ?k (G) and ?ek (G) as of Schreier graphs of Aut+ (Fk ). It was shown by Gilman [Gi] that Aut(Fk ) acts on ?ek (G) as AN or SN , where N = j?ek (G)j, provided that graph ?k (G) is connected and G is simple. Consider the case G = PSL(2; p). It is known that ?k (PSL(2; p)) is connected for k 3 [Gi]. Now, if the question 1) above has a positive answer, the Corollary 2 implies that f?k (p) = ?k (PSL(2; p))g is an expander, for k 8. On the other hand, Gilman's result (see above) shows that a quotient graph ?ek (p) = ?k (p)=GL(2; p) is a Schreier graph of AN or SN , each of them in nitely often. Therefore, if 4) has a negative answer, then f?ek (p)g cannot be an expander, which contradicts 1). In a di erent direction, since Aut(Fk ) is mapped onto SN , the positive answer to question 3) implies that for 4). Finally, let us show here that if ?bk (p) = ?k (PSL(2; p)N )=GL(2; p)N , where N = N (k; p) = j?k (PSL(2; p)j=jGL(2; p)j, are expanders for some xed k 3, then the positive answer to 4) follows. This is a weaker condition than 3) (see above). Indeed, Gilman's result implies that Aut+ (Fk ) acts transitively on ?bk (p) for in nitely many primes p. But in fact, Hall's result [Ha] (see also [KL]) gives that vertices in ?bk (p) are exactly permutations of all vertices in ?ek (p). Therefore, ?bk (p) is a Cayley graph of SN . This implies the claim.
Acknowledgements We are grateful to Persi Diaconis, Alex Lubotzky, and Peter Sarnak, for their profound in uence on our work. We thank Noga Alon, Andrew Odlyzko, Prasad Tetali, and Jan Saxl for help with the references.
2
Transpositions and inversions of elements are the remaining Nielsen generators [MKS]
EXPANSION OF GRAPHS
15
References [AM] N. Alon, V. D. Milman, 1 , isoperimetric inequalities for graphs, and superconcentrators, J. Comb. Theory, Ser. B 38 (1985), 73-88. [AR] N. Alon, Y. Roichman, Random Cayley graphs and expanders, Random Structures and Algorithms 5 (1994), 271{284. [B1] L. Babai, The probability of generating the symmetric group, J. Comb. Th. Ser. A 52 (1989), 148{153. [B2] L. Babai, Local expansion of vertex-transitive graphs and random generation in nite groups, in Proc. 23rd ACM STOC, 1991, 164{174. [B3] L. Babai, Automorphism groups, isomorphism, reconstruction, in Handbook of Combinatorics (R. L. Graham, M. Groetschel, and L. Lovasz, eds.), Elsevier, 1996. [B4] L. Babai, Randomization in group algorithms: Conceptual questions, in Groups and Computation II (L. Finkelstein, W.M. Kantor, eds.) DIMACS Workshops on Groups and Computation, AMS, Providence, 1997. [BH] L. Babai, G. L. Hetyei, On the diameter of random Cayley graphs of the symmetric group, Combin. Probab. Comput. 1 (1992), 201{208. [B+] L. Babai, G. Hetyei, W. M. Kantor, A. Lubotzky, A . Seress, On the diameter of nite groups, Proc. 31st IEEE FOCS, 1990, 857{865. [BKL] L. Babai, W. M. Kantor, A. Lubotzky, Small-diameter Cayley graphs for nite simple groups, European J. Combin. 10 (1989), 507{522. [BP] L. Babai, I. Pak, Strong bias of group generators: an obstacle to the \product replacement algorithm", Proc. 11th ACM-SIAM Symposium on Discrete Algorithms, 2000, 627{635. [Bo] A. Borel, Linear Algebraic Groups, (Second edition), Springer, 1991. [C+] F. Celler, C. R. Leedham-Green, S. Murray, A. Niemeyer, E. A. O'Brien, Generating random elements of a nite group, Comm. Algebra 23 (1995), 4931{4948. [Ch] F. R. K. Chung, Spectral Graph Theory, AMS, Providence, RI, 1994. [CG] F. R. K. Chung, R. L. Graham, Random walks on generating sets for nite groups, Electronic J. of Comb. 4:2 (1997), #R7. [CT] F. R. K. Chung, P. Tetali, Isoperimetric inequalities for cartesian products of graphs, Combin. Probab. Comput. 7 (1998), 141{148. [CV] M. Culler, K. Vogtmann, A group theoretic criterion for property FA, Proc. Amer. Math. Soc 124 (1996), 677-683. [DaS] G. Davido , P. Sarnak, An elementary approach to Ramanujan graphs unpublished monograph. [Di] P. Diaconis, Group Representations in Probability and Statistics, IMS, Hayward, California, 1988. [DS1] P. Diaconis, L. Salo {Coste, Comarison techniques for random walk on nite groups, Annals of Probability 21 (1993), 2131{2156. [DS2] P. Diaconis, L. Salo -Coste, Walks on generating sets of abelian groups, Probability Theory and Related Fields 105 (1996), 393{421. [DS3] P. Diaconis, L. Salo -Coste, Walks on generating sets of groups, Invent. Math. 134 (1998), 251{199. [Dx] J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969), 199{205. [GG] O. Gabber, Z. Galil, Explicit constructions of linear size superconcentrators, Proc. 20th IEEE FOCS, 1979, 364{370. [Ga] A. Gamburd, Spectral gap for in nite index \congruence" subgroups, Israel Journal of Mathematics, to appear. [GJS] A. Gamburd, D. Jakobson, P. Sarnak, Spectra of elements in the group ring of SU(2), J. European Math. Soc. 1 (1999), 51{85. [Gi] R. Gilman, Finite quotients of the automorphism group of a free group, Canad. J. Math. 29 (1977), 541{551. [G1] M. Gromov, Hyperbolic groups, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987. [G2] M. Gromov, Spaces and Questions, GAFA (2000 Special Volume, Part I), 118{161.
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[HR] H. Halberstam, H.-E. Richert, Sieve Methods, LMS Monographs, Academic Press, London, 1974. [Ha] P. Hall, The Eulerian functions of a group, Quart. J. Math. 7 (1936), 134{151. [HT] C. Houdre, P. Tetali, Isoperimetric invariants for product Markov chains and graph products, preprint, Georgia Tech, 1996. [JS] M. Jerrum, A. Sinclair, Approximating the permanent, SIAM J. Comput. 18 (1989), 1149{ 1178. [K1] W. M. Kantor, Some topics in asymptotic group theory, in Groups, Combinatorics & Geometry (Durham, 1990), Cambridge Univ. Press, Cambridge, 1992, pp. 403{421. [K2] W. M. Kantor, Simple groups in computational group theory, in Proc. ICM Berlin, Vol. II (1998), 77{86. [KL] W. M. Kantor, A. Lubotzky, The probability of generating a nite classical group, Geom. Dedicata 36 (1990), 67{87. [LR1] J. La erty, D. Rockmore, Fast Fourier analysis for SL2 over a nite eld and related numerical experiments, Experimental Mathematics 1 (1992), 115-139. [LR2] J. La erty, D. Rockmore, Numerical investigation of the spectrum for certain families of Cayley graphs, DIMACS Series in Disc. Math. and Theor. Comp. Sci. 10 (1993), 63-73. [LG] C. R. Leedham{Green, personal communication. [LS1] M. W. Liebeck, A. Shalev, The probability of generating a nite simple group, Geom. Dedicata 56 (1995), 103{113. [LS2] M. W. Liebeck, A. Shalev, Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky, J. Algebra 184 (1996), 31{57. [Lu] A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Birkhauser, Boston, 1994. [LP] A. Lubotzky, I. Pak, The product replacement algorithm and Kazhdan's property (T), Journal of AMS 52 (2000), 5525{5561. [LPS] A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan graphs., Combinatorica 8 (1988), 261{277. [LW] A. Lubotzky, B. Weiss, Groups and expanders, in Expanding graphs (Princeton, NJ, 1992), 95{109, DIMACS Ser., 10, AMS, Providence, RI, 1993. [MKS] W. Magnus, A. Karrass, D. Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations (Second edition), Dover, New York, 1976. [M1] G. A. Margulis, Explicit constructions of expanders, Problems of Information Transmission 9 (1973), 325{332. [M2] G. A. Margulis, Explicit group theoretic constructions of combinatorial schemes, and their applications for construction of expanders and concentrators, Problems of Information Transmission 24 (1988), 39{46. [Mc] J. McCool, A faithful polynomial representation of Out F3 , Math. Proc. Cambridge Philos. Soc. 106 (1989), 207{213. [O] A. Odlyzko, personal communication. [P1] I. Pak, Random walks on nite groups with few random generators, Electr. J. Prob. 4 (1999), 1{11. [P2] I. Pak, The probability of generating a nite group, preprint (1999). [P3] I. Pak, What do we know about the product replacement algorithm?, in \Groups and Computation III" (W. Kantor, A. Seress, eds.), Berlin, 2000, 301{347. [P4] I. Pak, The product replacement algorithm is polynomial, Proc. 41st IEEE FOCS, 2000. [PB] I. Pak, S. Bratus, On sampling generating sets of nite groups and the product replacement algorithm, Proc. 11th ISSAC, 1999, 91{96. [PZ] I. Pak, A. Zuk, Two Kazhdan constants and mixing of random walks, preprint (2001). [Q] J.-J. Quisquater, unpublished manuscript. [RVW] O. Reingold, S. Vadhan, A. Wigderson, Entropy Waves, The Zig-Zag Graph Product, and New Constant-Degree Expanders and Extractors, Proc. 41st IEEE FOCS, 2000. [S1] Y. Shalom, Expanding graphs and invariant means, Combinatorica 17 (1997), 555-575. [S2] Y. Shalom, Expander graphs and amenable quotients, in: Emerging applications of number theory, 571{581, IMA Vol. Math. Appl., 109, Springer, New York, 1999. [S3] Y. Shalom, Bounded generation and Kazhdan's property (T) 90 (1999), Publ. Math. IHES, 145{168.
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B. Sury, Bounded generation does not imply nite presentation, Comm. Algebra 25 (1997), 1673{1683. [W1] J. Whiston, Maximal independent generating sets of the symmetric group, J. Algebra 232 (2000), 255{268. [W2] J. Whiston, Ph.D. Thesis, Cambridge University, 2001.
MSRI Berkeley, CA 94720 E-mail: [email protected] Igor Pak
Department of Mathematics MIT, Cambridge, MA 02139 E-mail: [email protected] March 25, 2001 Abstract. We establish a connection between the expansion coecient of the product replacement graph ?k (G) and the minimal expansion coecient of a Cayley graph of G In particular, we show that the product replacement graphs ? with k generators. ?k PSL(2; p) form an expander family, under assumption that all Cayley graphs of PSL(2; p), with at most k generators are expanders. This gives a new explanation of the outstanding performance of the product replacement algorithm and supports the speculation that all product replacement graphs are expanders [LP,P3].
Introduction Expanders are highly connected sparse graphs of great interest in computer science, in areas ranging from parallel computation to complexity theory and cryptography; recently they were also used as a key ingredient in connection with the Baum-Connes conjecture [G2] and in computational group theory [LP]. The explicit constructions of expander graphs (by Margulis [M1, M2] and Lubotzky, Phillips, and Sarnak [LPS] ) use deep tools (Kazhdan's property (T), Selberg's Theorem, proved Ramanujan conjectures) to construct families of Cayley graphs of nite groups. The fundamental problem, raised by Lubotzky and Weiss [LW], is whether being an expander family is a property of the groups alone, independent of the choice of generators (Independence Problem). The product replacement algorithm is a commonly used heuristic to generate random group elements in a nite group. Let G be a nite group generated by at most d elements. The product replacement graph ?k (G) is de ned to be a graph, with vertices corresponding to generating k-tuples in G, and edges corresponding to Nielsen transformations. While ?k (G) is closely related to Cayley graphs of G, Key words and phrases. Random walks, mixing time, expander graphs, Independence Problem.
Typeset by AMS-TEX 1
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ALEXANDER GAMBURD, IGOR PAK
these graphs can be de ned as the Schreier graphs of a special group automorphisms of a free group Aut+ (Fk ). Most recently, graphs ?k (G) became a subject of an intense investigation, prompted by the study of a commonly used `practical' algorithm for generating random elements in nite groups, designed Leedham-Green and Soicher [LG]. This algorithm, based on the random walk on product replacement graphs, showed a remarkable performance, as reported in [C et al.] It was suggested in [LP], and, in fact, proved in several special cases, that the product replacement graphs ?k (G) are expanders, for a xed k, when jGj ! 1. The main result of this paper is a theorem, establishing the connection between the expansion coecient of the product replacement graph ?k (G) and the minimal expansion coecient of a Cayley graph of G with k generators. In particular, we show that if one assumes that all Cayley graphs with at most four generators in PSL(2; p) have a universal lower bound on expansion, then the product replacement graphs ?k (PSL(2; p)) form an expander family, when k 8 is xed, and p ! 1. Let ? be a k-regular (oriented) graph with an adjacency matrix A. For the rest of the paper we assume that ? is symmetric, i.e. that A = AT . Consider a nearest neighbor random walk W = W (?), with transition matrix P = A=k. Denote by 1 = 0 > 1 2 : : : the eigenvalues of P, and let (?) = 1 ? 1 be the eigenvalue gap of the graph ?. We say that a sequence of k-regular graphs f?ng is an expander family, if for some > 0, we have (?n ) > , for all n 1. Among many properties of expanders are the bounds on the isoperimetric constant (see below), diameter of the graph diam(?n ) C1 log j?n j, and the mixing time of the random walk mix W (?n ) C2 log j?n j, for some universal constants C1 ; C2 > 0. Let G be a nite group, and let S be a generating set with k elements. We will always assume that S is symmetric : S = S ?1 . Denote by C = C (G; S ) the corresponding Cayley graph on G. Consider ? a nearest neighbor random walk W (G; S ) = W (C ). Denote by (G; S ) = C the eigenvalue gap of C (G; S ). As in case of general graphs, we say that a family of Cayley graphs fCn = C (Gn ; Sn )g is an expander, if there exist " > 0, such that (Cn ) > " for all n. For a xed integer m, we say that a sequence of groups fGn g has universal expansion with m generators, if there ? exist > 0, such that for every n and every hS i = Gn , jS j m, we have C (Gn ; Sn ) > 0. The positive answer to the Independence Problem for PSL(2; p) is the main assumption in this paper: 1) Does the sequence of groups fPSL(2; p); p ? primeg have universal expansion with m = 4 generators? 1 An armative answer to problem 1 is supported by numerical experiments [LR1,LR2] and some recent results [S1,Ga]; see comments in section 6. Let us de ne the product replacement graph ?k (G) as follows. Let vertices be all generating k-tuples of the group G, and let edges correspond to transformations : Li;j and Ri;j : (g1 ; : : : ; gi ; : : : ; gk ) ! (g1 ; : : : ; gi g1 ; : : : ; gk ); Ri;j j 1 Li;j : (g1 ; : : : ; gi ; : : : ; gk ) ! (g1 ; : : : ; gj gi ; : : : ; gk ): 1 In can be shown, in fact, that if 1) holds, then fPSL(2; p)g has universal expansion for every xed m 4 (cf. section 6.)
EXPANSION OF GRAPHS
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These graphs were introduced in [C et al.], in connection with computing in nite groups. Note that graphs ?k (G) are regular, of degree D = 4 k (k ? 1). Let fGn g be a sequence of nite groups, generated by at most d elements, and let k d be xed. As before, we say ? that a sequence f?k (Gn )g is an expander family, if for some > 0 we have ?k (Gn ) > for all n. The main question of our study is the following open problem: ? 2) Does the sequence of graphs ?k PSL(2; p) ; p ? prime form an expander family , for any xed k 3 ? We prove that a positive answer to question 1) implies a positive answer to question 2), for all k 8. In fact, we prove a general result, for every nite group G. We show that, under certain conditions, the Cheeger constant of ?k (G) is bounded from below by the minimal Cheeger constant of the Cayley graph C (G; S ), with jS j k. This idea is similar in spirit to the paper [DS3], where the eigenvalue gap (?k (G)) was bounded in terms of maximal diameter of C (G; S ) (cf. [P3]). For k = (log jGj), the dependence on diameter was later removed in [P4]. Let us say a few words about the proof. The proof is combinatorial in nature and is almost entirely self contained. At the end, we rely upon some results on the group structure of PSL(2; p), which are known in the literature (see below). We use a novel combinatorial technique based on graph decomposition, as opposed to path arguments used in previously in [CG,DS2,DS3,P4]. It is easy to see that such a technique can never prove that a certain family of graphs is an expander family (cf. [P3]). The rest of the paper is structured as follows. In section 1 we state the main results of the paper. Preliminary observations and lemmas are presented in sections 2, and 3. These follow with the proof of main results (section 4) and proof of the lemmas (section 5). We conclude in section 6 with a collection of historical and mathematical remarks, and pointers to the literature. Throughout the paper, [n] will denote f1; 2; : : :; ng. We use G to denote a nite group, and ? to denote a connected regular graph. 1. Main results
Let G be a nite group, d = d(G) be the minimal number of generators of G. We say that the set of generators S is minimal, if no proper subset of S generates G. By m(G) denote the maximal size of the minimal generating set of G. By `(G) denote the length of the maximal subgroup chain of G. Clearly, d(G) m(G) `(G) log2 jGj: Let 'k (G) denotes the probability that k random group elements generate G. Let (G) be the smallest k such that 'k (G) > 1 ? . It was shown in [P1] that (G) `(G) + C log(1=), for a universal constant C > 0. Let ? be an (oriented, loops are allowed) graph. Denote by deg(?) the maximal in and out-degree of a vertex in ?. We say that ? is k-regular if every vertex has in and out-degree k = deg(?). De ne (edge) expansion e(?) as follows: ) ( E? (X; X ) j ? j e(?) = min k jX j : X ?; 1 jX j 2 ;
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ALEXANDER GAMBURD, IGOR PAK
where E? (X; Y ) = f(x; y) 2 ? : x 2 X; Y 2 Y g is the set of edges between X and Y , and k = deg(?). Note that 1 > e(?) > 0. The Cheeger-Buser inequality (in this context, also known as conductance bound of Jerrum and Sinclair [JS]) gives:
e(?) (?) e(?) 8 : 2
Thus, a uniform lower bound on expansion e(?n ) > > 0, for a family of k-regular graphs f?n g, is an equivalent de nition of expanders [Lu]. Let C (G; S ) be an (oriented) Cayley graph on G, with a generating set S . Denote by k (G) the smallest expansion of the Cayley graph on G with at most k generators:
?
k (G) = min e C (G; S ) : hS i = G; jS j k : Let ?k (G) be the product replacement graph, de ned as in the introduction Let D = deg(?k (G)) = 4 k (k ? 1).
Main Theorem Let G be a nite group. For every k 2 m(G), there exist = (k) > 0, such that if k 2 (G), then ? e ?k (G) > c k (G); where c = c(k) is a constant, which depends only on k, and not on G. I Note that the result in the theorem holds for every nite group G, not a family of groups. Recall that for any sequence fGn g of simple groups, with jGn j ! 1, we have '2 (G) ! 1, as n ! 1 (see section 2 below). Therefore, for every such sequence fGn g, and > 0, we have (Gn ) ! 2, as n ! 1. The following corollaries follow from Main Theorem.
Corollary 1. Let fGng be a family of nite simple groups, such that jGn j ! 1, as n ! 1. Suppose also that m(Gn ) m, and m (Gn ) > 0, for all n 1. Let k 2 m, D = 4k(k ? 1). Then a family of D-regular graphs ?k (Gn ) is an expander family. I Corollary 2. Let k 8 be a xed integer, and let D = 4k(k ? 1). Assume that there exists > 0, such that ? 4(PSL(2 ; p)) , for all prime p 5. Then a family of D-regular graphs ?k PSL(2; p) is an expander family. I Corollary 3. Let fGng be a family of nite groups, such that `(Gn ) `, for all n ! 1. Suppose also that ` (Gn ) > 0, for all n 1. There exists a universal constant C > 0, such that for all k 2 ` + C log `, the family of 4k(k ? 1)-regular graphs ?k (Gn ) is an expander family. I The Main Theorem and the corollaries will be proved in section 4.
Remark 1. The product replacement graphs of simple groups, studied in Corollary 1, seem to complement the set of graphs ?k (G) that are known to be
EXPANSION OF GRAPHS
5
expanders. Indeed, the only other cases, when ?k (G) are shown to be expanders, are the abelian groups and nilpotent groups of bounded nilpotency class [LP]. But in these cases the Cayley graphs have large diameter and cannot be expanders (see [B+] and section 6 below.) Also, although Corollary 3 is stated in general terms, it can, in fact, be applied to variety of algebraic groups (see section 2 below.) 2. Combinatorics and probability on finite groups
Let G be a nite group, and let
'k (G) = P(hg1 ; : : : ; gk i = G) = j?jkG(jGk )j be the probability that k independent random elements in G generate the whole
group. A major recent result in this direction was completed in a sequence of papers by Dixon [Dx] (see also [B1]), Kantor and Lubotzky [KL], Liebeck and Shalev [LS1,LS2]. Together, these papers prove that '2 (Gn ) ! 1, for any sequence of nite simple groups fGn g, such that jGn j ! 1. While the overall ? result is based on classi cation of nite simple groups, the special cases '2 PSL(2; p) ! 1 as p ! 1, and '2 (An ) ! 1 as n ! 1, are completely elementary. In our notation, # (PSL(2; p) = # (An ) = 2 for all > 0, and p, n large enough. Note, that if #1=2 (G) < r (i.e. 'r (G) < 1=2), we easily have 'k (Gn ) < , for k > C r log(1=), and for some universal constant C > 0 (see [P2]). In this case j?k (G)j > (1 ? )jGjk . It is also known that if ` = `(G), then for all k > ` + C log(1=) we have 'k (Gn ) < , for some universal constant C [P2]. While the bound m(G) `(G) is often sharp, there are examples when m(G) is much smaller than `(G) (see [W1,W2]). While the recent work [W2] calculates for a number of simple groups, we will use only result m(PSL(2; p)) 4. There is little doubt that all our results can be generalized to all series of bounded rank. Note that this condition is crucial, since we trivially have m(PSL(n; p)) n ? 1. Let us note here that `(G) is bounded for a large number of algebraic groups, which extends the Corollary 3 beyond simple groups. Indeed, for a series of algebraic groups fG(p)g of the same rank, over Fp (such as fPSL(n; p)g, when n is xed), the order f (p) = ord(G(p)) is a polynomial in p 3 of a xed degree n2 [Bo]. Thus, the sieve methods in number theory imply that f (p) has at most a bounded number of prime factors for in nitely many primes p (see [HR], chapter 8,9.) Therefore, `(Gp ) < C for in nitely many prime p, where C = C (n) is a xed constant. In particular, for G(p) = PSL(2; p), we have f (p) = ord(G(p)) = 21 p(p ? 1)(p + 1). It is believed [O] that there are in nitely many primes q, such that 6q + 1 and 12q +1 are also primes. Taking p = 12q +1, this gives f (p) = 12p(6p +1)(12p +1), so that `(PSL(2; p)) 6 for in nitely many primes p. On the other hand, one can deduce from [HR] that `(PSL(2; p)) 13 for in nitely many primes p. We will also need the following simple result, which we prove in section 5. Lemma 1. Let 1 > > > 0. Consider a nite group G, and let X G, such that 1 jX j (1 ? )jGj. Then P? g X ? X > jX j > 1 ? 11 ?? ;
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ALEXANDER GAMBURD, IGOR PAK
where g is uniform in G. I 3. Edge expansion of graphs
In this section we present some known and some new results on edge expansion of graphs. The lemmas are arranged so that the level of generality roughly increases. Since at no point we need sharp bounds, we do not attempt to optimize the constants. Instead, we present simple proofs of (sometimes, known) lemmas so that their generalization can be obtained with no diculty. Let ? = (V; E ) be a nite (oriented) graph. A graph ?0 = (V 0 ; E 0 ) is called a subgraph of ?, if V 0 V and E 0 E . Let ? be a k-regular graph, and let e(?) be the (edge) expansion of ?, de ned as above. It is often convenient to work with the Cheeger constant of G is de ned to be h(?) = e(?) k.
Lemma 2. Let ? = (V; E ) be a nite k-regular graph, ?1 = (V1 ; E1 ), : : : , ?n = (Vn ; En ) be the subgraphs of ?, such that V = [i Vi , and jVi j > (1 ? )jV j, for some 0 < < 51 and all i 2 [n]. Then h(?) maxf1n; 5g min h(?i ) : i 2 [n] : I This lemma is probably well known, although we were unable to locate the precise reference. We postpone the proof until section 5.
Lemma 3. Let X [M ] [N ], jX j (M N=2). Denote Xi; = X \fig [N ], X;j = X \ [M ] fj g. Then, for some universal constants ; > 0, we have: (i; j ) 2 X : X < (1 ? ) N + (i; j ) 2 X : X < (1 ? ) M i; ;j (~) > jX j: Moreover, for all < we have: jf(i; j ) 2 X : Xi; < (1 ? ) N; i (1 ? ) M gj (~~) + jf(i; j ) 2 X : X;j < (1 ? ) M gj > ( ? ) jX j: I Versions of the rst part of Lemma 3 seem to be known, with roughly the same elementary proof. Since we need the second part as well, we present the proof of lemma for values = 1=10 and = 31=90. While these are probably not optimal, they suce for our purposes. For graphs ?1 = (V1 ; E1 ) and ?2 = (V2 ; E2 ), de ne the cartesian product ? = ?1 ?2 to be the graph ? = (V; E ), such that V = V1 V2 and
?
E = (v1 ; v2 ); (v10 ; v20 ) 2 V 2 : (v1 ; v10 ) 2 E1 ; v2 = v20 or (v2 ; v20 ) 2 E2 ; v1 = v10 :
EXPANSION OF GRAPHS
7
Let k1 = deg(?1 ) and k2 = deg(?2 ). Note that deg(?) = k1 + k2 .
Proposition 1. [CT,HT] Let ? = ?1 ?2 be the product of graphs ?1 and ?2 . Let h1 = h(?1 ), h2 = h(?2 ). Then h(?) 21 min h1 ; h2 : I The proof is elementary, and follows from Lemma 3 (perhaps, with a di erent constant.) As we need an extension of the proposition, we include a proof with a constant 1=27 instead of 1=2. Let us also quote, without a proof, a known generalization of this result:
Proposition 2. [CT,HT] Let ? = ?1 ?m be the product of graphs
?1 ; : : : ; ?m . Then
h(?) 21 min h(?i ) : i 2 [m] : I
We say that ?0 = (V 0 ; E 0 ) is a restriction of ? = (V; E ), if V 0 V , and (v1 ; v2 ) 2 E , v1 ; v2 2 V 0 , implies that (v1 ; v2 ) 2 E 0 . ? M ] [N ]; E ) be a k-regular graph. De ne ?i; = Vi; ; Ei; , ?;j = ?V Let; E? =,([with Vi; = fig [N ], V;j = [M ] fj g, to be restrictions of ?. ;j ;j
Lemma 4. There exist constants ; > 0, such that for all 0 < the following holds. Let ? = ([M ] [N ]; E ), k = deg(?). Consider restrictions ?i; and ?;j , de ned as above. De ne
?
h1 = min h ?;j : j 2 [N ] ; ? h2 = min h ?i; : i 2 [(1 ? )M ] : Then h(?) min fh1; h2 g. I
In section 5 we deduce the lemma from our proof of (a weaker version of) Proposition 1. Below we present a nal extension of Lemma 3, tailored to our needs. Let ?i = (Ei ; [N ]); i 2 [M ] be a family of k-regular graphs on [N ]. We say that f?i g has -uniform expansion with Cheeger constant bh, if for all X [N ], such that 1 jX j N=2, we have Ei (X; X ) bh2 jX j, for at least (1 ? ) M di erent i 2 [M ]. Of course, if h(?i ) bh for all i 2 [(1 ? )M ] (cf. Lemma 4), then f?i g has -uniform expansion with Cheeger constant bh.
Lemma 5. There exist constants ; > 0, such that for all 0 < the following holds. Let ? = ([M ] [N ]; E ), k = deg(?). Consider restrictions ?i; and ?;j , de ned as above. De ne
?
h1 = min h ?;j : j 2 [N ] ;
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ALEXANDER GAMBURD, IGOR PAK
and suppose ?i; ; i 2 [M ]g is a family of k-regular graphs, which has -uniform expansion with Cheeger constant bh2 . Then h(?) min fh1; bh2 g. I
Remark 2. Let us note that in Lemmas 3, 4 and 5, we cannot weaken the condition to have -error for both types of restrictions. For example, in Lemma 4, we cannot let j 2 [(1 ?)N ]. Similarly, we cannot allow to have -uniform expansion for a family ?i; as well. Indeed, in these cases there can be very small sets X ? which lie in the intersection of `bad' directions. This is the main reason why we cannot weaken our assumption 1) to a weaker version of it, with all Cayley graphs of PSL(2; p) substituted by random Cayley graphs. 4. Proof of the Main Theorem and Corollaries
Proof of Main Theorem.
Fix 1=2 > > 0, and let n = maxfm(G); (G)g, r = k ? n. Since k 2(G) and k 2 m(G), we obtain r maxf(G); m(G)g. De ne an action of Sk on ?k (G) as follows : (g1 ; : : : ; gk ) = (g(1) ; : : : ; g(k) ), for 2 Sk . Consider a subgraph ?0 with vertices all generating k-tuples (g1 ; : : : ; gk ) 2 ? = ?k (G), such that hg1 ; : : : ; gn i = G, and edges corresponding to transforma , L , such that 1 j n < i k, or 1 i n < j k. Consider also tions Ri;j i;j ? 0 ? , de ned as above, for each coset representative 2 (k; n) = Sk = Sn Sr . Clearly, ?0 ' ?0 for all 2 Sk . We have j?0 j > (1 ? ) j?j, since, by de nition, n (G). Also, for every (g) = (g1 ; : : : ; gk ) 2 ?k (G), there exists 0 2 (k; n), such that (g) 2 0 ?0 (this follows from n m(G)). From Lemma 2, we obtain:
h(?) ?1k min fh(?0 ) : 2 (k; n)g = C h(?0 ); n
for some constant C = C (n; k). Now let us prove that h(?0 ) > cn (G). Think of ?0 as a graph on ?n (G) Gr . For every xed? (g) = (g1 ; : : : ; gn) 2 ?n (G), consider ?0(g); ?0 , the subgraph of ?0 with vertices (g); (h) , where (h) 2 Gr is any r-tuple of elements. De ne ?0;(h) ?0 analogously, for every (h) 2 Gr . We have k0 = deg ?0;(h) = deg ?0(g); = 4 n r. De ne C (G; S ) = C (G; S ) [ C ?1 (G; S ) to be a union of two isomorphic Cayley corresponding to multiplication on the left and on the right: C = (g;graphs g s1 ); (g; s1?g) : g 2 G; s 2 S . Clearly, h(C ) = 2 h(C ). Now,? by de ni tion, ?0(g); = C G; fg1; : : : ; gn g r , and by Proposition 2, we have h ?0(g); 1 h(C ) > k 0 (G). n 2 ? We cannot prove that h ?0;(h) > > 0 for two reasons. First, not all elements (h) 2 Gr are generating sets (although there are > (1 ? )jGjr of them). The graphs ?;(h) are disconnected when (h) 2= ?r (G). Thus, we cannot conclude that Cayley graphs on these r-tuples are expanders. Second, graphs ?0;(h) =
?
EXPANSION OF GRAPHS
9
C G; fh1; : : : ; hr g n are not products of (union of) Cayley graphs C , but their
intersection with ?k (G). Thus we cannot use Proposition 2 to bound Cheeger constant. In fact, we cannot do this for any xed (h) 2 Gr . Instead, we use Lemma 5 to establish "-uniform expansion of the family ?0;(h) on ?n (G), for (h) 2 Gr . Indeed, consider rst H = Gn and any subset X H , 1 jX j jH j=2. Now apply Lemma 1 to the group H (taking = 1=2 and = 1=4). We obtain that the di erence in the lemma is > jX j=4, for > jX j=3 di erent g 2 H . Now observe that for uniform (h) 2 Gr , the rst n components (h)0 = (h1 ; : : : ; hn ) in (h) are uniform in H . Multiplication of (g) by (h)0 is a composition of transformations L1;n+1 L2;n+2 : : : Ln;2n . By the symmetry, if the composition has expansion > , at least one of the transformations Li;n+i has expansion > =n (cf. [B2]). Since j?n (G)j > (1 ? )jH j, this gives jg X ? X j \ ?n (G)j > (1=4n ? )jX j = jX j for at least jX j=3 di erent g 2 H . This proves the 1=3-uniform expansion for the family of graphs ?0;(h) , with Cheeger constant > . Now take = minf1=4 1=90; 1=8ng, so as to satisfy the lemmas. From Lemma 5 we conclude that h(?0 ) > C (n; k)n (G). Now the theorem follows from the observations above.
Proof of Corollary 1. Since '2 (Gn ) ! 1, and d(Gn ) = 2, the second condition k 2(Gn ) = 4 is trivial. The corollary now follows immediately from the Main Theorem. Proof of Corollary 2.
This is a special case of Corollary 1. Recall that
m(PSL(2; p)) 4, and the result follows.
Proof of Corollary 3. Recall that m(G) `(G), and '`+t = 1 ? O(1=2t) [P2]. Finally, observe that k (G) (k=`)` (this follows by removing extra edges). In the proof of Main Theorem we need to nd k and , such that '(k=2) 1 ? , and = O(1=k). Since `(Gn ) is bounded, we can solve these two equations by taking t = O(log k). We omit the easy calculation. 5. Proofs of Lemmas
Proof of Lemma 1. Note that E?jg X \ X j =
X
x;x 2X 0
Markov inequality gives
P(gx = x0) = jX j2 jG1 j (1 ? )jX j:
P?jg X ? X j < jX j = P?jg X \ X j > (1 ? )jX j < 11 ?? : This implies the result.
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ALEXANDER GAMBURD, IGOR PAK
Proof of Lemma 2. Let X V , 1 jX j jV j=2. Consider subsets Xi = X \ V . Denote Ei (X; Y ) = E? (X; Y ), for X; Y Vi , and let ei = e(?i ). Also, let ki = deg(?i ), k = deg(?). Fix a constant = (1 ? )=2 > 0. Note that 2 < < 1. 5 2 There are two possible cases. Either jX j jV j, or jX j > jV j. We consider them separately. In the rst case, jXi j < 1? jVi j = jVi j=2. Therefore jEi (Xi ; Vi ? Xi )j > ei jXi j ki . Since X [i Xi , there is always i 2 [n], such that jXi j jX j=n. Therefore, for this i we have: i
E? (X; X ) jEi (Xi ; Vi ? Xi )j (ei ki ) jXn j : We conclude:
E? (X; X ) 1 min e k : e(?) = X : 1jmin kn i i i X jjV j=2 k jX j In the second case, we have :
jXi j > ( ? ) jV j ( ? ) 2 jX j > 25 jX j;
jXi j jX j 21 jV j < 2(1 1? ) jVi j < 58 jVi j;
and therefore jVi ? Xi j=jXi j < 1?5=58=8 = 35 . For every i 2 [n], we have :
E? (X; X ) Ei (Xi ; Vi ? Xi ) ei ki min fjXi j; jVi ? Xi jg > 35 ei ki jXi j > 35 ei ki 25 jX j > 51 ei ki jX j: We conclude:
E? (X; X ) 1 min e k : e(?) = X : 1jmin X jjV j=2 k jX j 5k i i i This completes the second case and proves Lemma 2.
Proof of Lemma 3. Let = 1=10 and = 31=90. Denote I = i 2 [M ] : jXi; j < 109 N , J = j 2 [N ] : jX;j j < 109 M . Since jX j MN=2, we have 9 MN 10 N (M ? jI j) < jX j 2 ; which gives jI j > 49 M . Therefore, for every j 2 J , we have 31 jX;j ? I fj gj > 109 M ? 59 M = 31 90 M 90 X;j :
EXPANSION OF GRAPHS
Now
11
X X + X X ? I fj g ;j ;j j 2J j 2= J X > (1 ? ) jX j + 31 X;j = (1 ? ) jX j + 31 jX j 31 jX j;
P = [i2I Xi; + [j2J X;j = j 2= J 90
90
90
where P is equal to the l.h.s. of (~) in the lemma, and
=
P
jX;j j 0: jX j
j 2= J
This proves the rst part (~). The second part follows verbatim, except for a substitution of I by I 0 = I \ [(1 ? )M ], and the constant 31=90 is replaced by (31=90 ? ), as in (~~).
Proof of Proposition 1. Recall that we prove only a weaker version of the proposition, with constant 1=27 instead of 1=2, as in the claim. Suppose ?1 = ([M ]; E1 ), ?2 = ([N ]; E2 ). Let ?i; = fig ?2 , ?;j = ?1 fj g, for all i 2 [M ], j 2 [N ]. Let X ?, 1 jX j j?j=2. As in Lemma 3, let Xi; = ?i; \ X , X;j = ?;j \ X . Consider 9 N ; J = j 2 [N ] : X < 9 M : I = i 2 [M ] : Xi; < 10 ;j 10 Also, let
?
?
Ei; = E1 Xi; ; ?i; ? Xi; ; E;j = E2 X;j ; ?;j ? X;j : By de nition of I and J , for all i 2 I , j 2 J we have:
min Xi; ; ?i; ? Xi; > 19 Xi; ; min X;j ; ?;j ? X;j > 19 X;j : By Lemma 3, we have: E (X; ? ? X ) X E + X E X e k Xi; + X e k X;j i; ;j 2 2 1 1 9 9 i2I j 2J i2I j 2J 0 1 X X minfe1 k1 ; e2 k2 g @ Xi; + X;j A 9
We conclude:
i2I
j 2J
jX j min fe k ; e k g: j X j 19 min fe1 k1 ; e2 k2 g 31 1 1 2 2 90 27
e(?) = X : 1jmin X j MN=2
E (X; ? ? X ) k jX j
271 k min fe1 k1 ; e2 k2 g:
12
ALEXANDER GAMBURD, IGOR PAK
Proof of Lemma 4. Let = 1=27, and = 1=90. Use the second part of Lemma 3. Substitute = 901 to obtain that the r.h.s. of (~~) is at least 3190?1 = 13 . Now note that we never used in the proof of Proposition 1 the fact that ?i; (and, similarly, graphs ?;j ) are isomorphic to each other. Now the proof of the lemma follows verbatim the proof of Proposition 1, with the only di erence that we use (~~) instead of (~), with = 1=90, as above. Proof of Lemma 5. Follows verbatim the proof of Lemma 4. Indeed, notice again that in the proof of Proposition 1 we never used the fact that i is always in the same subset of size (1 ? )M in [M ]. Similarly, for every X [N ] we never used the full expansion of ?i; , but rather Ei; = Ei (X; X ). The rest of the proof remains unchanged. 6. Final Remarks
Let us elaborate on the rich history of the problem and known results, related to both questions 1) and 2) in the introduction. It is well known that, in a certain precise sense, \random" k-regular graphs are expanders. Only a much weaker result is known for Cayley graphs, when k is allowed to grow with jGj. The best known bound for all nite group is the case when k = (log jGj) [AR] (see also [P1]). While this bound cannot be improved for abelian groups, no better result is known for other classes of groups (cf. [B3]). The rst explicit constructions of expanders were found by Margulis [M1], who used Kazhdan's property (T) from representation theory to prove the expansion. The next breakthrough came in papers [LPS,M2], where the authors used harmonic analysis and number theory to obtain the explicit constructions of so called Ramanujan graphs, the expanders with the largest possible eigenvalue gap (when k is xed). Both approaches use Cayley graphs of linear groups, and neither of them is elementary, although an e ort to simplify the technique has been made (see [GG,Lu,DaS].) Most recently, combinatorial constructions of expanders has been introduced in [RVW]. One can think of our results as of new approach to develop expanders. In case of Cayley graphs, only very special generators has been used, although recent improvements increase the variety of such sets (see [Ga,GJS,S1,S2]). These results support an armative answer to the Independence Problem 1) for PSL(2; p), i.e. that all Cayley graphs of PSL(2; p) form an expander family (see [LW]). Further support is given by numerical evidence [LR1,LR2]. Our results indicate the importance of this problem for computational group theory. Let us note Independence Problem remains open even for \random" generating sets [B+,Lu], and there seem to be little hope of proving 1) with existing techniques. On the other hand, it was speculated in [LW] that universal expansion property must hold for all group sequences, which admit some expanding family of Cayley graphs. If true, this would allow us to prove expansion for a large family of product replacement graphs.
EXPANSION OF GRAPHS
13
As we mentioned above, the product replacement graphs ?k (G) in this form were introduced recently in connection with the `practical' algorithm for generating random elements [C et al.]. On the other hand, a related family of graphs ?ek (G) was studied back in the sixties by B.H. Neumann, M. Dunwoody, and others, in connection with the so called T-systems (see [P3] for the references). Many basic questions about these graphs remain unanswered, such as the connectivity of ?k (G), for general nite groups G. In our running example, it was proved by Gilman that graphs ?k (PSL(2; p)) are connected, for k 3, and p 5 [Gi]. In general, it is known that ?k (G) is connected for all k > m(G) + d(G) [DS3,P3]. Now, a rigorous study of convergence of random walks on the product replacement graphs ?k (G), for general nite groups G and in special cases, was undertaken in a number of recent papers [B4,CG,DS2,DS3,LP,P3]. In the latest paper [P4], the second author showed that the random walk mixes in time polynomial in k and log jGj, for k = (log jGj). Still, for small k, the nature of the practical rapid mixing remains unclear. One possible explanation came in [LP], where the authors showed that ?k (G) are always expanders, provided a known open problem 3) has positive solution: 3) Does group Aut(Fk ) have Kazhdan's property (T) ? The problem 3) remains open; an indirect evidence in favor of it is the fact proved in [CV] the it has property (FA) of Serre. It is also known that Aut(Fk ) are hyperbolic and thus nonamenable [G1]. There are also some negative indications: Aut(F2 ) and Aut(F3 ) are shown not to have (T) [Mc], and Aut(Fk ) do not have bounded generation [Su], a property closely related to (T) [S3]. Now, since the authors in [C+] test the product replacement algorithm on a number of simple and quasisimple groups, one can think of this work as an alternative explanation of the algorithm performance. Let us mention here that it was proved (unconditionally) in [LP], that graphs ?k (G) are expanders, when G is nilpotent of class `, and both k and ` are xed. It is entirely possible that any family of graphs ?k (G) , for a xed k, is an expander. While a counterexample to this claim would give a negative answer to 3), a proof of this would not, however, imply 3). We refer to an extensive survey article [P3] for references and details. Let us note that the main theorem is inapplicable to a family of alternating groups fAn g, where n 5. Not all Cayley graphs of An are expanders (see below), and also m(An ) = n ? 2 [W1], which contradicts the assumptions in Corollary 1. Let us present here an important closely related open problem [B+,Lu,LW]: 4) Is there any sequence of Cayley graphs fC (Sn ; Rn ) , which is an expander (for some generating sets hRn i = Sn .) Not unlike question 1), question 4) remains dicult if not unapproachable. Only recently, a sequence of bounded generating sets hRn i = Sn , with diam C (Sn ; Rn ) = O(n log n), has been constructed [BKL,Q]. It was widely speculated that the answer to 4) is negative, i.e. that there are no expanders on Sn [LW]. At the moment, not even generating sets with mix W = O(n log n) are known. The sets Rn , as above, come close with mix W (Sn ; Rn ) = O(n log3 n) [DS1]. To add to a confusion, let us mention here a conjecture that all Cayley graphs on Sn have diameter
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ALEXANDER GAMBURD, IGOR PAK
at most O(n2 ) [B3,Di], while for \random" Cayley graphs the diameter ?O(ispnbe-) lieved to? be O(n log n ) [K1]. The best bounds in both cases are exp and exp O(log2 n) , respectively [B+,BH]. It is easy to nd a non-expanding family in Sn , i.e. Rn = f(1; 2); (1; 2; : : :; n)1 g, such that diam C (Sn ; Rn ) = (n2 ) (see [Di,DS1,Lu]). Still, for all we know, \random" Cayley graphs on Sn can be expanders [B3,BH]. Let us conclude with an interesting observation in [LP], which connects all questions 1) { 4). First, consider a diagonal action of Aut(G), de ned as follows : (g1 ; : : : ; gk ) = ( (g1 ); : : : ; (gk )), for 2 Aut(G). De ne a graph ?ek (G) with vertices corresponding to orbits of action of Aut(G) and edges corresponding to (see [Gi,P3]). Clearly, if ?k (G) is connected, then ?ek (G) is also Li;j and Ri;j connected [P3]. as of Now, let Fk be a free group on k generators. One can think of Li;j and Ri;j 2 + (special ) Nielsen generators in Aut(Fk ). Let Aut (Fk ) = hLi;j ; Ri;j i Aut(Fk ). It is easy to see that Aut+ (Fk ) is a subgroup of index 2 in Aut(Fk ) [LP,P3]. One can think of graphs ?k (G) and ?ek (G) as of Schreier graphs of Aut+ (Fk ). It was shown by Gilman [Gi] that Aut(Fk ) acts on ?ek (G) as AN or SN , where N = j?ek (G)j, provided that graph ?k (G) is connected and G is simple. Consider the case G = PSL(2; p). It is known that ?k (PSL(2; p)) is connected for k 3 [Gi]. Now, if the question 1) above has a positive answer, the Corollary 2 implies that f?k (p) = ?k (PSL(2; p))g is an expander, for k 8. On the other hand, Gilman's result (see above) shows that a quotient graph ?ek (p) = ?k (p)=GL(2; p) is a Schreier graph of AN or SN , each of them in nitely often. Therefore, if 4) has a negative answer, then f?ek (p)g cannot be an expander, which contradicts 1). In a di erent direction, since Aut(Fk ) is mapped onto SN , the positive answer to question 3) implies that for 4). Finally, let us show here that if ?bk (p) = ?k (PSL(2; p)N )=GL(2; p)N , where N = N (k; p) = j?k (PSL(2; p)j=jGL(2; p)j, are expanders for some xed k 3, then the positive answer to 4) follows. This is a weaker condition than 3) (see above). Indeed, Gilman's result implies that Aut+ (Fk ) acts transitively on ?bk (p) for in nitely many primes p. But in fact, Hall's result [Ha] (see also [KL]) gives that vertices in ?bk (p) are exactly permutations of all vertices in ?ek (p). Therefore, ?bk (p) is a Cayley graph of SN . This implies the claim.
Acknowledgements We are grateful to Persi Diaconis, Alex Lubotzky, and Peter Sarnak, for their profound in uence on our work. We thank Noga Alon, Andrew Odlyzko, Prasad Tetali, and Jan Saxl for help with the references.
2
Transpositions and inversions of elements are the remaining Nielsen generators [MKS]
EXPANSION OF GRAPHS
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