Appears in collection : Groupes, géométrie et analyse : conférence en l'honneur des 60 ans d'Alain Valette

We provide an explicit construction of an infinite sequence of finite 4-regular Cayley graphs as in the title. For each dimensionn superior at 2, our graphs are suitable Cayley graphs of SL n(Fp) as primep→∞. These are the first explicit examples in all dimensions n superior at 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 arethe only known explicit large girth Cayley graph expanders with bounded diameter-by-girth ratio. This is a joint work with Arindam Biswas.