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An explicit presentation for asymptotically rigid mapping class groups

Sergio Domingo-Zubiaga

Abstract

We show that several families of asymptotically rigid mapping class groups arise as explicit quotients of the fundamental group of a graph of groups, with mapping class groups as vertex and edge stabilizers. Using this description, and building on the work of Labruère and Paris, we compute explicit presentations for asymptotically rigid mapping class groups of surfaces.

An explicit presentation for asymptotically rigid mapping class groups

Abstract

We show that several families of asymptotically rigid mapping class groups arise as explicit quotients of the fundamental group of a graph of groups, with mapping class groups as vertex and edge stabilizers. Using this description, and building on the work of Labruère and Paris, we compute explicit presentations for asymptotically rigid mapping class groups of surfaces.

Paper Structure

This paper contains 37 sections, 29 theorems, 89 equations, 30 figures.

Table of Contents

  1. Statement of results.
  2. Preliminaries
  3. Boundary permuting mapping class group
  4. Tree surfaces and rigid structures
  5. Asymptotically rigid mapping class groups
  6. Stein-Farley cube complex
  7. Graphs of groups and fundamental groups
  8. Proof of Theorem \ref{['Main1']}
  9. A cocompact action
  10. Constructing a fundamental domain
  11. Towards an explicit presentation for $\mathcal{B}$
  12. Vertex and edge stabilizers in the surface case
  13. Presentation of the boundary permuting mapping class group of surfaces
  14. Exact sequences and presentations
  15. Generators of the boundary permuting mapping class group
  16. ...and 22 more sections

Key Result

Theorem I

Let $\mathcal{B}$ be a group in one of the following families: Then $\mathcal{B}$ is isomorphic to an explicit quotient of the fundamental group of a graph of groups $\mathcal{G}_\mathcal{B}$ over a snake graph, where vertex groups are explicit finite extensions of mapping class groups of compact manifolds, and the extra relations correspond to identifying mapp

Figures (30)

  • Figure 1: A snake graph.
  • Figure 2: The tree surface $\mathcal{S}_{2,1}(O,Y)$, with $O$ of genus 3 and $Y$ of genus 1.
  • Figure 3: A tongued snake.
  • Figure 4: On the left, the manifolds defining $W_\mathcal{K}$ when $r\geq1$. On the right, the manifolds defining $W_\mathcal{K}$ when $r=1$.
  • Figure 5:
  • ...and 25 more figures

Theorems & Definitions (50)

  • Theorem I
  • Theorem II
  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['LemDeslink']}
  • ...and 40 more