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Notes on hyperelliptic mapping class groups

Marco Boggi

Abstract

Hyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural epimorphisms between mapping class groups of surfaces with marked points. We study these groups in a systematic way. An application of this theory is a counterexample to the genus $2$ case of a conjecture by Putman and Wieland on virtual linear representations of mapping class groups. In the last section, we study profinite completions of hyperelliptic mapping class groups: we extend the congruence subgroup property to the general class of hyperelliptic mapping class groups introduced above and then determine the centralizers of multitwists and of open subgroups in their profinite completions.

Notes on hyperelliptic mapping class groups

Abstract

Hyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural epimorphisms between mapping class groups of surfaces with marked points. We study these groups in a systematic way. An application of this theory is a counterexample to the genus case of a conjecture by Putman and Wieland on virtual linear representations of mapping class groups. In the last section, we study profinite completions of hyperelliptic mapping class groups: we extend the congruence subgroup property to the general class of hyperelliptic mapping class groups introduced above and then determine the centralizers of multitwists and of open subgroups in their profinite completions.

Paper Structure

This paper contains 41 sections, 39 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Hyperelliptic mapping class groups
  3. Surfaces
  4. Mapping class groups
  5. Mapping class groups of disconnected surfaces
  6. Relative mapping class groups
  7. Hyperelliptic surfaces
  8. Hyperelliptic mapping class groups
  9. Hyperelliptic mapping class groups of a marked hyperelliptic surface
  10. Hyperelliptic mapping class groups of disconnected surfaces
  11. Hyperelliptic mapping class groups of marked disconnected surfaces
  12. Relative hyperelliptic mapping class groups
  13. The complex of symmetric curves on $(S,{\upsilon})$
  14. Geometric interpretation of hyperelliptic mapping class groups
  15. Moduli stacks of complex curves of a fixed topological type
  16. ...and 26 more sections

Key Result

Theorem A

With the above notations, there is a natural short exact sequence: where $N_\circ$ is the normal subgroup generated by the Dehn twists about symmetric separating simple closed curves on $S^\circ$ bounding a disc which contains no marked points and only the two punctures obtained removing $Q$ and ${\upsilon}(Q)$.

Theorems & Definitions (78)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 68 more