2007 | OriginalPaper | Chapter
Macbeaths infinite series of Hurwitz groups
Author : Amir Džambić
Published in: Arithmetic and Geometry Around Hypergeometric Functions
Publisher: Birkhäuser Basel
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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 PSL
2
(
$$ \mathbb{F}_7 $$
), 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.