Genus actions of finite simple groups (Q1108371)

For integers r,s,t\(\geq 2\), consider a tessellation of the plane P by triangles having angles \(\pi\) /r, \(\pi\) /s, and \(\pi\) /t (P spherical, euclidean, or hyperbolic). Let T(r,s,t) be the group of symmetries of P preserving this tessellation. To each epimorphism \(\eta\) : T(r,s,t)\(\to G\), there corresponds a conformal action of G on the Riemann surface \(S=P/\ker \eta\). For G simple nonabelian, we show, with one exception, that G is normal in Aut(S). We further show that Aut(S) embeds in Aut G, provided G is (2,m,n)-generated and S is any surface of least genus on which G acts. Finally, for such G and S, we extend the normality result to surfaces arising from epimorphisms of the form \(\eta\) : \(\Gamma\) \(\to G\), where \(\Gamma\) is arbitrary Fuchsian.