Logarithmic girth expander graphs of $SL_n(\mathbb F_p)$

We provide an explicit construction of finite 4-regular graphs $(\Gamma_k)_{k\in \mathbb N}$ with ${girth \Gamma_k\to\infty}$ as $k\to\infty$ and $\frac{diam \Gamma_k}{girth \Gamma_k}\leqslant D$ 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 $p\to\infty$ 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.
Comment: Title and content updated to reflect published version. Previous title: "Large girth graphs with bounded diameter-by-girth ratio"