Classification of automorphism groups, up to topological equivalence, of compact Riemann surfaces of genus 4
Let \((X,G)\) be a pair of a compact Riemann surface of genus \(g>1\) and a group of automorphisms of \(X\). A pair \((X_1,G_1)\) is said to be topologically equivalent to \((X_2,G_2)\) if there exist an orientation preserving homeomorphism \(h:X_1 \to X_2\) and an isomorphism \(\iota :G_1 \to G_2\) such that \(\iota(\sigma)\circ h=h\circ \sigma\) for every \(\sigma \in G_1\). In the case \(g=4\), the author classifies \((X,G)\)'s up to topological equivalence and shows that there are 75 topological equivalence classes. In order to classify \((X,G)\)'s, the author considers surface kernel homomorphisms \(\varphi:\Gamma \to G\) and classifies them up to topological equivalence by using the works of \textit{I. Kuribayashi} and \textit{A. Kuribayashi} [J. Pure Appl. Algebra 65, No. 3, 277-292 (1990; Zbl 0709.30036)] on the matrix representation of Aut\((X)\) and branching data.
scientific article; zbMATH DE number 1955946