We provide an explicit construction of finite 4-regular graphs (Γk)k∈N\documentclass[12pt]{minimal}
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\begin{document}$$(\Gamma _k)_{k\in {\mathbb {N}}}$$\end{document} with girthΓk→∞\documentclass[12pt]{minimal}
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\begin{document}$$\text {girth}\, \Gamma _k\rightarrow \infty $$\end{document} as k→∞\documentclass[12pt]{minimal}
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\begin{document}$$k\rightarrow \infty $$\end{document} and diamΓkgirthΓk⩽D\documentclass[12pt]{minimal}
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\begin{document}$$\frac{\text {diam}\,\Gamma _k}{\text {girth}\,\Gamma _k}\leqslant D$$\end{document} for some D>0\documentclass[12pt]{minimal}
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\begin{document}$$D>0$$\end{document} and all k∈N\documentclass[12pt]{minimal}
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\begin{document}$$k\in {\mathbb {N}}$$\end{document}. For each fixed dimension n⩾2,\documentclass[12pt]{minimal}
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\begin{document}$$n\geqslant 2,$$\end{document} we find a pair of matrices in SLn(Z)\documentclass[12pt]{minimal}
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\begin{document}$$SL_{n}({\mathbb {Z}})$$\end{document} such that (i) they generate a free subgroup, (ii) their reductions modp\documentclass[12pt]{minimal}
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\begin{document}$$\bmod \, p$$\end{document} generate SLn(Fp)\documentclass[12pt]{minimal}
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\begin{document}$$SL_{n}({\mathbb {F}}_{p})$$\end{document} for all sufficiently large primes p, (iii) the corresponding Cayley graphs of SLn(Fp)\documentclass[12pt]{minimal}
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\begin{document}$$SL_{n}({\mathbb {F}}_{p})$$\end{document} have girth at least cnlogp\documentclass[12pt]{minimal}
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\begin{document}$$c_n\log p$$\end{document} for some cn>0\documentclass[12pt]{minimal}
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\begin{document}$$c_n>0$$\end{document}. 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(logp)\documentclass[12pt]{minimal}
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\begin{document}$$O(\log p)$$\end{document}. This gives infinite sequences of finite 4-regular Cayley graphs of SLn(Fp)\documentclass[12pt]{minimal}
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\begin{document}$$SL_n({\mathbb {F}}_p)$$\end{document} as p→∞\documentclass[12pt]{minimal}
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\begin{document}$$p\rightarrow \infty $$\end{document} with large girth and bounded diameter-by-girth ratio. These are the first explicit examples in all dimensions n⩾2\documentclass[12pt]{minimal}
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\begin{document}$$n\geqslant 2$$\end{document} (all prior examples were in n=2\documentclass[12pt]{minimal}
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\begin{document}$$n=2$$\end{document}). 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.
