Simple Hurwitz groups and eta invariant

Abstract

A Hurwitz group is a conformal automorphism group of a compact Riemann surface with precisely $84(g - 1)$ automorphisms, where $g$ is the genus of the surface. Our starting point is a result on the smallest Hurwitz group $\mathrm{PSL}(2,\mathbb{F}_{7})$ which is the automorphism group of the Klein surface. In this paper, we generalize it to various classes of simple Hurwitz groups and discuss a relationship between the surface symmetry and spectral asymmetry for compact Riemann surfaces. To be more precise, we show that the reducibility of an element of a simple Hurwitz group is equivalent to the vanishing of the $\eta$-invariant of the corresponding mapping torus. Several wide classes of simple Hurwitz groups which include the alternating group, the Chevalley group and the Monster, which is the largest sporadic simple group, satisfy our main theorem.