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Mapping class groupoids and Thompson's groups

A. Muhammed Uludağ, Mustafa Topkara, Özge Ülkem, Ayberk Zeytin

Abstract

We introduce a groupoid ${\mathbf{ΠMG}}}$, called the fundamental modular groupoid, which is a variant of Penner's mapping class groupoid. We study how it relates to the surface mapping class groups and Thompson's group $\mathsf T$. We also introduce larger groupoid $\mathbf{ΩMG}$, which is related to outer automorphisms of free groups and Thompson's group $\mathsf V$ in a similar manner.

Mapping class groupoids and Thompson's groups

Abstract

We introduce a groupoid , called the fundamental modular groupoid, which is a variant of Penner's mapping class groupoid. We study how it relates to the surface mapping class groups and Thompson's group . We also introduce larger groupoid , which is related to outer automorphisms of free groups and Thompson's group in a similar manner.

Paper Structure

This paper contains 26 sections, 18 theorems, 83 equations, 29 figures.

Table of Contents

  1. Keywords
  2. Introduction
  3. Notation.
  4. Two versions of Mapping Class Groupoids
  5. Dual picture: spines
  6. Flips
  7. Flips of spines with a d.o.e.
  8. Fundamental modular groupoid of a surface
  9. Combinatorial graphs as non-Hausdorff spaces
  10. A variation of Penner's mapping class groupoid
  11. Fundamental groupoids of modular graphs
  12. Introduction to the fundamental modular groupoid
  13. Punctures
  14. Full fundamental modular groupoid
  15. Question.
  16. ...and 11 more sections

Key Result

Lemma \oldthetheorem

(Whitehead decorated, see also hatcher.) Any two elements of ${\mathrm{TRN}}(S)$ are connected via a finite sequence of flips. Equivalently, any two elements of ${\mathrm{SPN}}(S)$ are connected via a finite sequence of flips.

Figures (29)

  • Figure 1: How to choose the d.o.e. in the graph dual to a triangulation. Unbroken arcs constitute the triangulation, dashed lines constitute the dual spine.
  • Figure 2: A triangulation flip and its effect on the dual trivalent ribbon graph spine (shown by dashed lines). Hollow nodes denotes punctures, and thick lines denotes ideal arcs.
  • Figure 3: A degenerate triangulation and its dual spine.
  • Figure 4: At each step, the flip is applied on the highlighted edge. After five flips, the twin edges represented by $e$ and $f$ are transposed.
  • Figure 5: A trivalent graph and the corresponding bipartite graph.
  • ...and 24 more figures

Theorems & Definitions (40)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 30 more