Advancing a decades-old problem in extremal graph theory

For every prime power q we construct an explicit infinite family of (q³ + 1)-regular graphs whose girth satisfies girth ≥ 85 log_q³ n − O(1), where n is the number of vertices. This improves the constant 43 of Margulis and of Lubotzky, Phillips and Sarnak, which has been the best known since 1988. The graphs are quotients of the Bruhat-Tits tree of a special unitary group in four variables of Witt index one, over an unramified quadratic extension of a local field with residue field 𝔽_q. We divide the tree by congruence subgroups of an arithmetic group that comes from a division algebra of degree four with a positive involution of the second kind. The girth bound rests on a certificate: for a nontrivial congruence element w of level 𝔞, the number det(w − I) is a nonzero element of a totally real field, divisible by 𝔞⁴, whose size at the place of the tree is controlled by the distance that w moves vertices of the tree. For one explicit family of 28-regular graphs we also prove a matching upper bound, so that there the girth equals 85 log₂₇ n + O(1).