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Large Girth Graphs with Bounded Diameter-By-Girth Ratio 3

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Large Girth Graphs with Bounded Diameter-By-Girth Ratio 3 LARGE GIRTH GRAPHS WITH BOUNDED DIAMETER-BY-GIRTH RATIO GOULNARA ARZHANTSEVA AND ARINDAM BISWAS Abstract. We provide an explicit construction of finite 4-regular graphs (Γk)k∈N with girth Γk k → ∞ as k and diam Γ 6 D for some D > 0 and all k N. For each dimension n > 2, we find a → ∞ girth Γk ∈ pair of matrices in SLn(Z) such that (i) they generate a free subgroup, (ii) their reductions mod p generate SLn(Fp) for all sufficiently large primes p, (iii) the corresponding Cayley graphs of SLn(Fp) have girth at least cn log p for some cn > 0. Relying on growth results (with no use of expansion properties of the involved graphs), we observe that the diameter of those Cayley graphs is at most O(log p). This gives infinite sequences of finite 4-regular Cayley graphs as in the title. These are the first explicit examples in all dimensions n > 2 (all prior examples were in n = 2). Moreover, they happen to be expanders. Together with Margulis’ and Lubotzky-Phillips-Sarnak’s classical con- structions, these new graphs are the only known explicit large girth Cayley graph expanders with bounded diameter-by-girth ratio. 1. Introduction The girth of a graph is the edge-length of its shortest non-trivial cycle (it is assigned to be infinity for an acyclic graph). The diameter of a graph is the greatest edge-length distance between any pair of its vertices. We regard a graph Γ as a sequence of its connected components Γ = (Γk)k∈N each of which is endowed with the edge-length distance. Definition 1.1 (large girth graph, cf. [Big98]). A graph Γ=(Γk)k∈N is large girth if girth Γk , and it is logarithmic girth if there exists a constant c> 0 such that for all k N: →∞ ∈ girth Γ > c log Γ . k | k| Definition 1.2 (dg-bounded graph [AT18]). A graph Γ =(Γk)k∈N is dg-bounded if there exists a constant D> 0 such that for all k N: ∈ diam Γ k 6 D. girth Γk > 1 > A dg-bounded graph with uniformly bounded degree r satisfies girth Γk D diam Γk and diam Γk > > 1 arXiv:1803.09229v3 [math.GR] 27 Jan 2019 logr( Γk 1), for r 3, while diam Γk 2 ( Γk 1) for r = 2. Hence, for such a graph, girth Γk , whenever| |− Γ as k . | |− →∞ | k|→∞ →∞ In this paper, we focus on large girth dg-bounded graphs with uniformly bounded degree r > 2. For r = 2, a growing sequence of cycle graphs is an easy example of such a graph. A major theoretical and practical challenge is to build large girth dg-bounded graphs for r > 3. The existence of r-regular large girth dg-bounded graphs Γ = (Γk)k∈N, for each r > 3, is a classical result. For instance, first, one can show the existence of large girth graphs using the probabilistic argument of Erd˝os-Sachs [ES63] (see also [Sac63] for a recursive, on the degree r, construction) or by taking iterated mod 2-covers of a given graph (each iteration doubles the girth) and then, 2010 Mathematics Subject Classification. 20G40, 05C25, 20E26, 20F65. Key words and phrases. Large girth graphs, expander graphs, diameter, special linear group, coarse embedding, thin matrix group. This research was partially supported by the European Research Council (ERC) grant of Goulnara Arzhantseva, “ANALYTIC” grant agreement no. 259527. 1 2 GOULNARAARZHANTSEVAANDARINDAMBISWAS following Biggs [Big98, Lemma 3.1], prove that the diameter of the graph with the minimal number of vertices among all graphs of a given degree r > 2 and of girth at least g > 3 is at most g (giving D = 1 for such a graph, called a ‘cage’). However, this does not provide concrete examples and the resulting graphs are not necessarily Cayley graphs. The first explicit examples for r > 3 are given by suitable Cayley graphs of SL2(Fp), where Fp = Z/pZ are the integers mod p for a prime p. Namely, the famous explicit construction of large girth graphs by Margulis [Mar82] is made of 4-regular Cayley graphs Γ = Cay(SL (F ), A , B ) p 2 p { p p} with matrices Ap, Bp being images of Sanov’s generators of the free group (see below for notation and Section 3 for details). It satisfies girth Γp > C log Γp for C > 0 and it is dg-bounded as by Selberg’s theorem they form an infinite expander family| as p| and the Expander mixing lemma implies that diam Γ = O(log Γ ). Alternatively, the celebrated→∞ Lubotzky-Phillips-Sarnak [LPS88] p | p| graphs, Γq = Cay(P GL2(Fq), S1,S2,...,Sp+1 ) as q , where p, q are distinct primes congruent to 1 mod 4 with the Legendre{ symbol p = }1 and →∞S ,...,S are suitable p + 1 matrices, form q − 1 p+1 the Ramanujan family (hence, an expander family) and they satisfy girth Γq > 4 logp q logp 4 with 2 − Γq = q(q 1). Therefore, they are (p + 1)-regular large girth dg-bounded Cayley graphs. Observe that| | both of− these examples are in dimension 2, they are logarithmic girth, and the expansion property is used to conclude the dg-boundedness. Moreover, these constructions of Margulis and Lubotzky-Phillips-Sarnak (and their slight variants, again in dimension 2) have been, up to now, the only known explicit large girth dg-bounded expanders among the Cayley graphs. Our aim is to provide an explicit construction of 4-regular large girth dg-bounded Cayley graphs of SL (F ) as p , for all dimensions n > 2. Concretely, we shall give an extensive description n p →∞ of two-element generating sets of SLn(Fp) that induce the required properties of the Cayley graphs. We formulate now our main result, indicate several corollaries (among others, constructions of 2k-regular examples for all k > 2) and explain the motivation for this work. Magic matrices in dimension n > 2: 1 a 0 0 ... 0 1 0 0 ... 0 0 1 a 0 ... 0 b 1 0 ... 0 0 0 1 a . 0 0 b 1 ... 0 Let A = . and B = SLn(Z) with a, b > 2. 0 0 b . 0 ∈ . ... a . . 0000 ... 1 0 0 0 ... b 1 We denote by A and B their reductions modulo a prime p and by A, B 6 SL (Z) and A , B 6 p p h i n h p pi SLn(Fp) the subgroups generated by these matrices. Main Theorem. The matrices A and B satisfy the following. I. Freeness: For n =2, A, B is free; • h i For n =3, l > 4, Al, Bl is free; • ∀ h i For each n > 4, l > 3(n 1), Al, Bl is free. • ∀ − h i II. Generation mod p: For n =2, p prime with a, b 0(mod p), we have A , B = SL (F ); • ∀ 6≡ h p pi 2 p k N l l F For n = 3, a 1(mod 3), b 1(mod 3), l = 4 ,k , we have Ap, Bp = SL3( p) for • p =3 and all≡ primes p>K,≡ where − K is an∀ absolute constant;∈ h i LARGE GIRTH GRAPHS WITH BOUNDED DIAMETER-BY-GIRTH RATIO 3 For each n > 4, q prime with n 1(mod q) and a, b with a 1(mod q), b 1(mod q), • l 1 qk+1 +∀ 1 : k N,k > t, ≡with t N given by qt 6 n <≡ qt+1 , we have≡ Al , Bl = ∀ ∈{ }∪{ ∈ ∈ } h p pi SLn(Fp) for p = q and all primes p > L, where L is an absolute constant. III. Girth and diameter: For each of the following choices of the parameters: if n =2, l =1, then p prime with a, b 0(mod p); • ∀ 6≡ if n = 3, a 1(mod3), b 1(mod3), l = 4k,k N, then for p = 3 and all primes • p>K, where≡K is an absolute≡ − constant; ∀ ∈ if n > 4, q prime with n 1(mod q), and a, b with a 1(mod q), b 1(mod q), l • qk+1 +1 if∀ q =2 and qk+2 +1≡ if q = 2 : k N,k > t, with≡ t N given by≡ qt 6 n < q∀t+1 ∈, then{ for p = q6 and all primes p > L, where ∈L is an absolute constant,∈ } we have girth Cay(SL (F ), Al , Bl ) > c log p and diam Cay(SL (F ), Al , Bl )= O(log p), n p { p p} n n p { p p} where cn is an absolute constant. n,l F l l In particular, Γp (a, b) = Cay(SLn( p), Ap, Bp ) as p is a large girth dg-bounded graph, whenever n and l are as above. { } → ∞ Moreover, it is an expander. In addition, all the above absolute constants are effective. Taking specific dimension n and power l of our matrices gives, for instance, the following sequences of 4-regular large girth dg-bounded graphs: 1 4 0 4 1 0 0 4 3,4 F Γp (4, 2) = Cay(SL3( p), 0 1 4 , 2 1 0 ) as p { 0 0 1! 0 2 1! } →∞ and 10 10 1 40 0 1 0 0 0 0 14 0 7 1 0 0 Γ4,10(4, 4) = Cay(SL (F ), , ) as p . p 4 p {0 01 4 0 7 1 0 } →∞ 0 00 1 0 0 7 1 The graphs produced by the Main Theorem and the proof itself provide explicit examples relevant to various subjects. For instance, in each dimension n > 3, every free subgroup Al, Bl from Main n,l h i Theorem is an example of a thin matrix group.
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