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Automorphisms of smooth fine curve graphs

Katherine Williams Booth

Abstract

In this paper, we consider the automorphisms of fine curve graphs restricted to continuously $k$-differentiable curves. We show that for closed surfaces with genus at least 2, they are induced by homeomorphisms of the surface.

Automorphisms of smooth fine curve graphs

Abstract

In this paper, we consider the automorphisms of fine curve graphs restricted to continuously -differentiable curves. We show that for closed surfaces with genus at least 2, they are induced by homeomorphisms of the surface.

Paper Structure

This paper contains 18 sections, 42 theorems, 14 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Overview of the proof and paper outline
  4. Acknowledgments
  5. Constructing $\mathbf{\boldsymbol{\rho}: \mathop{\mathrm{\textbf{Aut}}}\nolimits \boldsymbol{\mathcal{EC}}^{\boldsymbol{\dagger}}\!\!\left(S; C^{k}\right) \boldsymbol{\rightarrow} \mathop{\mathrm{\textbf{Homeo}^{\textbf{k}}}}\nolimits(S)}$
  6. Distinguishing pairs of curves
  7. Separating, homotopic, and hulls of curves
  8. Recognizing subsurfaces with $\mathbf{\boldsymbol{\mathcal{C}}^{\boldsymbol{\dagger}}\!\!\left(S; C^{k}\right)}$
  9. Torus pairs
  10. $\mathbf{k}$-Smooth pairs
  11. Simple $\mathbf{k}$-smooth pairs
  12. Constructing $\mathbf{ \boldsymbol{\xi}^{-1}: \mathop{\mathrm{\textbf{Aut}}}\nolimits \boldsymbol{\mathcal{C}}^{\boldsymbol{\dagger}}\!\!\left(S; C^{k}\right) \boldsymbol{\rightarrow} \mathop{\mathrm{\textbf{Aut}}}\nolimits \boldsymbol{\mathcal{EC}}^{\boldsymbol{\dagger}}\!\!\left(S; C^{k}\right)}$
  13. Paired $k$-smooth arc graphs are connected
  14. Kernels of simple $\mathbf{k}$-smooth pairs
  15. Inessential curves
  16. ...and 3 more sections

Key Result

Proposition 2.1

Let $S$ be a smooth surface. Then there is a natural map such that for any $\alpha \in \mathop{\mathrm{Aut}}\nolimits \mathcal{EC}^{\dagger}\!\!\left(S; C^{k}\right)$, $(\widehat{\eta} \circ \rho)(\alpha) = \alpha$, where $\widehat{\eta}$ is the natural map from $\mathop{\mathrm{Homeo}}\nolimits^k(S)$ to $\mathop{\mathrm{Aut}}\nolimits \mathcal{EC}^{\dagg

Figures (26)

  • Figure 1: Bigon pair (left) and $k$-smooth pair (right)
  • Figure 2: Finding a curve that intersects the tail of $(c_i)$ near the point $x$
  • Figure 3: Pairs of separating (left) and non-separating (right) homotopic curves
  • Figure 4: The jointly separating case
  • Figure 5: A torus pair
  • ...and 21 more figures

Theorems & Definitions (83)

  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • proof : Proof of Proposition \ref{['ec1tohomeo1']}
  • ...and 73 more