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Automorphisms of the compression body graph

Ian Biringer, Nicholas G. Vlamis

TL;DR

When $S$ is a closed, orientable surface with genus g(S) \geq 2, it is shown that the automorphism group of the compression body graph $\mathcal{CB} (S)$ is the mapping class group.

Abstract

When $S$ is a closed, orientable surface with genus $g(S) \geq 2$, we show that the automorphism group of the compression body graph $\mathcal{CB}(S)$ is the mapping class group. Here, vertices are compression bodies with exterior boundary $S$, and edges connect pairs of compression bodies where one contains the other.

Automorphisms of the compression body graph

TL;DR

When is a closed, orientable surface with genus g(S) \geq 2, it is shown that the automorphism group of the compression body graph is the mapping class group.

Abstract

When is a closed, orientable surface with genus , we show that the automorphism group of the compression body graph is the mapping class group. Here, vertices are compression bodies with exterior boundary , and edges connect pairs of compression bodies where one contains the other.

Paper Structure

This paper contains 13 sections, 26 theorems, 34 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Sketch of the proof of Theorem \ref{['main']}
  3. Acknowledgements
  4. Compression bodies
  5. Small compression bodies
  6. Height of a compression body
  7. Separating and non-separating compression bodies
  8. Small compression bodies are invariant
  9. Links in $\mathcal{CB}(S)$
  10. Automorphisms preserve height: the proof of Proposition \ref{['prop:preservation']}.
  11. The torus complex
  12. The proof of Proposition \ref{['prop:intersection']}
  13. Automorphisms as mapping classes

Key Result

Theorem 1.1

When $g(S)\geq 2$, the natural map $\mathop{\mathrm{Mod}}\nolimits(S) \longrightarrow \mathop{\mathrm{Aut}}\nolimits(\mathcal{CB}(S))$ is a surjection.

Figures (8)

  • Figure 1: A boundary connected sum of four balls and interval bundles over a torus and a genus two surface. Here, the boundary connected sum of the four balls is a genus 2 handlebody.
  • Figure 2: The curve $a$ is drawn in red, and after an isotopy to make it disjoint from $a$, the curve $\partial D$ is drawn in blue. In both cases, the annulus $A$ bounded by $a$ and $\partial D$ starts from the left of $a$.
  • Figure 3: An $S$-compression body that has $g(S)$ interior boundary components, all of which are tori. A compressing system is drawn in red, and the interior boundary is blue. All such compression bodies are isomorphic, although not all compressing systems look like the above.
  • Figure 4: A handlebody with spots $D_1, D_2, D_3$ corresponding to the attachment of 1-handles. The non-separating meridians shown on the surface give a pants decomposition of the punctured surface obtained by deleting the spots from the boundary of the handlebody.
  • Figure 5: The two curves above are both meridians in only a single handlebody, while there are infinitely many separating curves that are disjoint from both.
  • ...and 3 more figures

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • proof
  • Corollary 2.4: Exterior-to-interior gluings
  • proof
  • Definition 2.5
  • ...and 38 more