Macbeaths infinite series of Hurwitz groups

Abstract

In the present paper we will construct an infinite series of so-called Hurwitz groups. One possible way to describe Hurwitz groups is to define them as finite homomorphic images of the Fuchsian triangle group with the signature (2, 3, 7). A reason why Hurwitz groups are interesting lies in the fact, that precisely these groups occur as the automorphism groups of compact Riemann surfaces of genus g > 1, which attain the upper bound 84(g − 1) for the order of the automorphism group. For a long time the only known Hurwitz group was the special linear group PSL2(\( \mathbb{F}_7 \)), with 168 elements, discovered by F. Klein in 1879, which is the automorphism group of the famous Kleinian quartic. In 1967 Macbeath found an infinite series of Hurwitz groups using group theoretic methods. In this paper we will give an alternative arithmetic construction of this series.KeywordsHurwitz groupsarithmetic Fuchsian groupsFuchsian triangle groups