Logarithmic girth expander graphs of $$SL_n({\mathbb {F}}_p)$$

Abstract

We provide an explicit construction of finite 4-regular graphs $$(\Gamma _k)_{k\in {\mathbb {N}}}$$ with $$\text {girth}\, \Gamma _k\rightarrow \infty $$ as $$k\rightarrow \infty $$ and $$\frac{\text {diam}\,\Gamma _k}{\text {girth}\,\Gamma _k}\leqslant D$$ for some $$D>0$$ and all $$k\in {\mathbb {N}}$$ . For each fixed dimension $$n\geqslant 2,$$ we find a pair of matrices in $$SL_{n}({\mathbb {Z}})$$ such that (i) they generate a free subgroup, (ii) their reductions $$\bmod \, p$$ generate $$SL_{n}({\mathbb {F}}_{p})$$ for all sufficiently large primes p, (iii) the corresponding Cayley graphs of $$SL_{n}({\mathbb {F}}_{p})$$ 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 $$SL_n({\mathbb {F}}_p)$$ as $$p\rightarrow \infty $$ with large girth and bounded diameter-by-girth ratio. These are the first explicit examples in all dimensions $$n\geqslant 2$$ (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.