Automorphism groups of q-trigonal planar Klein surfaces and maximal surfaces

Abstract

A compact Klein surface $X=\mathit{D}/\Gamma$, where $\mathit{D}$ denotes the hyperbolic plane and $\Gamma$ is a surface NEC group, is said to be $q$-trigonal if it admits an automorphism $\varphi$ of order $3$ such that the quotient has algebraic genus $q$. In this paper we obtain for each $q$ the automorphism groups of $q$-trigonal planar Klein surfaces, that is surfaces of topological genus $0$ with $k\ge 3$ boundary components. We also study the surfaces in this family, which have an automorphism group of maximal order (maximal surfaces). It will be done from an algebraic and geometrical point of view.