Seminár z algebraickej teórie grafov - Roman Nedela (29.11.2024)
v piatok 29.11.2024 o 13:15 hod. v posluchárni M VIII aj online
Prednášajúci: Roman Nedela (University of West Bohemia, Plzeň, Czech Republic)
Názov: Topological equivalence between groups of symmetries of Riemann surfaces and generation of the automorphism group of a Fuchsian group
Termín: 29.11.2024, 13:15 hod., M VIII a MS Teams
Abstrakt:
In my talk I will discuss the problem of classification of classes of topological equivalence of finite group actions on Riemann surfaces. By the Riemann-Hurwitz bound, there are just finitely many groups that act conformally on a closed orientable surface S_g of genus g≥2. With each such action of a group G on S_g one can associate the fundamental group Γ =π(O) of the quotient orbifold O=S_g/G, isomorphic to a Fuchsian group determined completely by the orbifold’s signature. The Riemann existence theorem reduces the problem of the existence of an action of G on S_g to a purely group-theoretical problem of deciding whether there is an smooth epimorphism mapping the Fuchsian group Γonto the group G. Using computer algebra systems such as Magma or GAP, together with the library of small groups, the generation of all finite group actions on a surface of fixed small genus g≥2 becomes almost a routine procedure. The difficult part is to determine the classes of these actions with respect to topological equivalence. To achieve this, one needs to investigate the action of the automorphism group of a Fuchsian group on the set of finite group actions on S_g with the corresponding signature. We derive several results on the topological equivalence of finite group actions on Riemann surfaces. As an application, we derive complete lists of finite group actions of genus g≤8 distinguished up to the topological equivalence. By a classical result of Zieshang every automorphism of a Fuchsian group lifts to an automorphism of the corresponding free group. Therefore the problem of determining the equivalence classes is closely related to the problem of determining a proper generating set of the group of lifts. The latter requires to employ techniques of combinatorial group theory.
Those of you who are not able to attend in person or who are still uncertain about the safety of attending in person are welcome to attend via MS Teams. In either case, we hope to see as many of you as possible (either in person or virtually) at our Friday gatherings.