Logarithmic girth expander graphs of SLn(Fp)

We provide an explicit construction of finite 4-regular graphs ( Γ k ) k ∈ N with girth Γ k → ∞ as k → ∞ and diam Γ k girth Γ k ⩽ D for some D > 0 and all k ∈ N . For each fixed dimension n ⩾ 2 , we find a pair of matrices in S L n ( Z ) such that (i) they generate a free subgroup, (ii) their reductions mod p generate S L n ( F p ) for all sufficiently large primes p , (iii) the corresponding Cayley graphs of S L n ( F p ) have girth at least c n log p for some c n > 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 of S L n ( F p ) as p → ∞ with large girth and bounded diameter-by-girth ratio. 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 constructions, these new graphs are the only known explicit logarithmic girth Cayley graph expanders.