Generating the liftable mapping class groups of cyclic covers of spheres

Kapari, Pankaj; Rajeevsarathy, Kashyap; Sanghi, Apeksha

doi:10.2140/agt.2025.25.2883

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Generating the liftable mapping class groups of cyclic covers of spheres

Pankaj Kapari, Kashyap Rajeevsarathy and Apeksha Sanghi

Algebraic & Geometric Topology 25 (2025) 2883–2903

DOI: 10.2140/agt.2025.25.2883

Abstract

For $g \geq 2$ , let $Mod (S_{g})$ be the mapping class group of a closed orientable surface $S_{g}$ of genus $g$ . We derive a finite generating set for the liftable mapping class groups corresponding to finite-sheeted regular branched cyclic covers of spheres. As an application, we provide an algorithm to derive presentations of these liftable mapping class groups, and the normalizers and centralizers of periodic mapping classes corresponding to these covers. Furthermore, we determine the isomorphism classes of the normalizers of irreducible periodic mapping classes in $Mod (S_{g})$ . Moreover, we derive presentations for the liftable mapping class groups corresponding to covers induced by certain reducible periodic mapping classes. Consequently, we derive a presentation for the centralizer and normalizer of a reducible periodic mapping class in $Mod (S_{g})$ of the highest order $2 g + 2$ . As final applications of our results, we recover the generating sets of the liftable mapping class groups of the hyperelliptic cover obtained by Birman and Hilden and the balanced superelliptic cover obtained by Ghaswala and Winarski.

Keywords

surface, periodic mapping class, normalizer, centralizer, liftable mapping class group

Mathematical Subject Classification

Primary: 57K20

Secondary: 57M60

References

Publication

Received: 10 September 2023

Revised: 28 March 2024

Accepted: 19 August 2024

Published: 17 September 2025

Authors

Pankaj Kapari
Department of Mathematics Indian Institute of Science Education and Research Bhopal Bhopal India

Kashyap Rajeevsarathy
Department of Mathematics Indian Institute of Science Education and Research Bhopal Bhopal India

Apeksha Sanghi
Department of Mathematics Indian Institute of Science Education and Research Mohali Mohali India

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Volume 25, issue 5 (2025)