Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Automorphisms of procongruence curve and pants complexes

Marco Boggi, Louis Funar

Abstract

In this paper we study the automorphism group of the procongruence mapping class group through its action on the associated procongruence curve and pants complexes. Our main result is a rigidity theorem for the procongruence completion of the pants complex. As an application we prove that moduli stacks of smooth algebraic curves satisfy a weak anabelian property in the procongruence setting.

Automorphisms of procongruence curve and pants complexes

Abstract

In this paper we study the automorphism group of the procongruence mapping class group through its action on the associated procongruence curve and pants complexes. Our main result is a rigidity theorem for the procongruence completion of the pants complex. As an application we prove that moduli stacks of smooth algebraic curves satisfy a weak anabelian property in the procongruence setting.

Paper Structure

This paper contains 38 sections, 55 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Rigidity of curve complexes
  4. Automorphisms of the curve complex
  5. Automorphisms of the pants graph
  6. The procongruence mapping class group and procongruence curve complex
  7. Profinite groups and the procongruence completion of the mapping class group
  8. Pro-completions of $G$-simplicial sets
  9. Automorphisms of simplicial profinite sets
  10. Congruence completions of curve complexes
  11. Profinite simple closed curves
  12. The procongruence curve complex
  13. Profinite Dehn twists in ${\check\Gamma}(S)$
  14. Profinite braid twists in ${\check\Gamma}(S)$
  15. The center of ${\check\Gamma}(S)$
  16. ...and 23 more sections

Key Result

Theorem A

For $S\neq S_{1,2}$ a connected hyperbolic surface such that $d(S)>1$, there is an exact sequence: where $O(S)$ is the finite set of the topological types of $(d(S)-1)$-multicurves on $S$. For $S$ of type $(1,2)$, the group $\operatorname{Aut}({\check C}_P(S_{1,2}))$ must be replaced with the subgroup of those automorphisms preserving the set of separating curves.

Theorems & Definitions (116)

  • Theorem A
  • Theorem B
  • Remark \oldthetheorem
  • Theorem C
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • ...and 106 more