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Graphs of curves on infinite-type surfaces with mapping class group actions

Matthew Gentry Durham, Federica Fanoni, Nicholas G. Vlamis

Abstract

We study when the mapping class group of an infinite-type surface $S$ admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on $S$. We introduce a topological invariant for infinite-type surfaces that determines in many cases whether there is such an action. This allows us to conclude that, as non-locally compact topological groups, many big mapping class groups have nontrivial coarse geometry in the sense of Rosendal.

Graphs of curves on infinite-type surfaces with mapping class group actions

Abstract

We study when the mapping class group of an infinite-type surface admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on . We introduce a topological invariant for infinite-type surfaces that determines in many cases whether there is such an action. This allows us to conclude that, as non-locally compact topological groups, many big mapping class groups have nontrivial coarse geometry in the sense of Rosendal.

Paper Structure

This paper contains 18 sections, 32 theorems, 19 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation and context
  3. Discussion of results
  4. Outline
  5. Conventions and standard definitions
  6. Background on infinite-type surfaces
  7. Graphs with finite-diameter mapping class group orbits
  8. Theorem \ref{['thm:main']} for finite-type surfaces
  9. Arcs
  10. Curves
  11. Non-singletons
  12. Quasi-retracts
  13. Quasi-retracts
  14. Witnesses
  15. Theorem \ref{['thm:main']} for infinite-type surface
  16. ...and 3 more sections

Key Result

Corollary 1.3

If $\mathop{\mathrm{\mathfrak{f}}}\nolimits(S) \geq 4$, then $\mathrm{MCG}(S)$ does not have property (OB).

Figures (5)

  • Figure 1: Dehn twists about the curves shown generate the pure mapping class group $\mathrm{PMCG}(F)$. The crosses denote punctures.
  • Figure 2: The subsurface $F$ is an ideal witness for $\mathop{\mathrm{Sep}}\nolimits_2(S,\mathop{\mathrm{Ends}}\nolimits(S))$.
  • Figure 3: The spotted Loch Ness monster surface (left) and the tripod surface (right).
  • Figure 4: The unbounded orbit of $\mathcal{G}'(S)$ on the once-punctured Jacob's ladder surface $S$
  • Figure 5: A geodesic between $a$ and $b$ in $\mathcal{G}(S)$.

Theorems & Definitions (73)

  • Remark 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 1
  • Theorem 2
  • Corollary 1.7
  • Theorem 3: AramayonaGeometry
  • ...and 63 more