TY - JOUR

T1 - Extending Harvey’s surface kernel maps

AU - Gilman, Jane

N1 - Funding Information:
2020 Mathematics Subject Classification. Primary 20H10; 32G15; Secondary 30F10; 30F35. Key words and phrases. conformal automorphism, Reidemeister-Schreier rewriting system, adapted generating sets, Riemann surfaces, mapping-class group, finite subgroups. Some of this work was carried out while the author was a supported Visiting Fellow at Princeton University or was supported by the Rutgers Research Council.
Publisher Copyright:
© 2022 American Mathematical Society.

PY - 2022

Y1 - 2022

N2 - Let S be a compact Riemann surface and G a group of conformal automorphisms of S with S0 = S/G. S is a finite regular branched cover of S0 . If U denotes the unit disc, let Γ and Γ0 be the Fuchsian groups with S = U/Γ and S0 = U/Γ0 . There is a group homomorphism of Γ0 onto G with kernel Γ and this is termed a surface kernel map. Two surface kernel maps are equivalent if they differ by an automorphism of Γ0 . In his 1971 paper Harvey showed that when G is a cyclic group, there is a unique simplest representative for this equivalence class. His result has played an important role in establishing subsequent results about conformal automorphism groups of surfaces. We extend his result to some surface kernel maps onto arbitrary finite groups. These can be used along with the Schreier-Reidemeister Theory to find a set of generators for Γ and the action of G as an outer automorphism group on the fundamental group of S putting the action on the fundamental group and the induced action on homology into a relatively simple format. As an example we compute generators for the fundamental group and an integral homology basis together with the action of G when G is S3, the symmetric group on three letters. The action of G shows that the homology basis found is not an adapted homology basis.

AB - Let S be a compact Riemann surface and G a group of conformal automorphisms of S with S0 = S/G. S is a finite regular branched cover of S0 . If U denotes the unit disc, let Γ and Γ0 be the Fuchsian groups with S = U/Γ and S0 = U/Γ0 . There is a group homomorphism of Γ0 onto G with kernel Γ and this is termed a surface kernel map. Two surface kernel maps are equivalent if they differ by an automorphism of Γ0 . In his 1971 paper Harvey showed that when G is a cyclic group, there is a unique simplest representative for this equivalence class. His result has played an important role in establishing subsequent results about conformal automorphism groups of surfaces. We extend his result to some surface kernel maps onto arbitrary finite groups. These can be used along with the Schreier-Reidemeister Theory to find a set of generators for Γ and the action of G as an outer automorphism group on the fundamental group of S putting the action on the fundamental group and the induced action on homology into a relatively simple format. As an example we compute generators for the fundamental group and an integral homology basis together with the action of G when G is S3, the symmetric group on three letters. The action of G shows that the homology basis found is not an adapted homology basis.

KW - Adapted generating sets

KW - Conformal automorphism

KW - Finite subgroups

KW - Mapping-class group

KW - Reidemeister-Schreier rewriting system

KW - Riemann surfaces

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U2 - 10.1090/conm/776/15607

DO - 10.1090/conm/776/15607

M3 - Article

AN - SCOPUS:85129406292

VL - 776

SP - 69

EP - 82

JO - Contemporary Mathematics

JF - Contemporary Mathematics

SN - 0271-4132

ER -