Algebraic Graph Theory Seminar - Ján Karabáš (20.3.2026)
Friday 20.13.2026 at 13:15, Lecture room M IX (online too)
Ján Karabáš (Slovak University of Technology):
Actions of finite groups on orientable surfaces
Abstract:
The problem of classifying all finite group actions on orientable surfaces of low genera is considered. We aim to extend the existing classifications (genera 2 and 3 by Broughton (1990), genus 4 by Bogopolsky (1991) and genus 5 by Kuribayashi and Kimura (1990)) to higher genera using computer algebra systems. For example, Marston Conder classified actions of all large groups (of orders ≥12(g-1)) on orientable surfaces of genera up to g≤101 (late 2000’s). These are the groups of automorphisms of maps and hypermaps of given genera. However, our goal is to go further: we aim to classify actions of all finite groups, focus on the topological equivalence of actions and its transfer to purely algebraic form, and, further, on the problem of implementing an algorithm to compute the classes of this equivalence. As follows, the problem of equivalence of group actions involves (at least partial) knowledge of the group of automorphisms of a given Fuchsian group — a deep and interesting problem itself. Recently (2024, Karabáš, Nedela, and Skyvová), we gave the complete classification of actions of finite groups up to genus 9.
Joint work with Domenico Catalano, Roman Nedela, and Mária Skyvová