Regular graphs of large girth and arbitrary degree

For every integer d≥10, we construct infinite families {G_n}_n∈ℕ of d+1-regular graphs which have a large girth ≥log_d|G_n|, and for d large enough ≥1.33 · log_d|G_n|. These are Cayley graphs on PGL₂(F_q) for a special set of d+1 generators whose choice is related to the arithmetic of integral quaternions. These graphs are inspired by the Ramanujan graphs of Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is a prime. When d is not equal to the power of an odd prime, this improves the previous construction of Imrich in 1984 where he obtained infinite families {I_n}_n∈ℕ of d + 1-regular graphs, realized as Cayley graphs on SL₂(F_q), and which are displaying a girth ≥0.48·log_d|I_n|. And when d is equal to a power of 2, this improves a construction by Morgenstern in 1994 where certain families {M_n}_n∈N of 2^k +1-regular graphs were shown to have girth ≥2/3·log₂^k|M_n|.