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On deformation spaces of quadratic rational maps

Tanya Firsova, Jeremy Kahn, Nikita Selinger

Abstract

We study the group of self-equivalences of a partially postcritically finite branched cover and answer a question of Adam Epstein about contractibility of certain deformation spaces of rational maps.

On deformation spaces of quadratic rational maps

Abstract

We study the group of self-equivalences of a partially postcritically finite branched cover and answer a question of Adam Epstein about contractibility of certain deformation spaces of rational maps.

Paper Structure

This paper contains 21 sections, 32 theorems, 51 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Background. Notations.
  4. Deformation spaces
  5. Supgroups of mapping class groups naturally associated to deformation spaces
  6. Postcritically finite branched covers
  7. Self-equivalences for partially postcritically finite maps
  8. Rees component
  9. Quadratic rational map
  10. Equalizing multicurves
  11. Invariant multicurves for postcritically finite maps
  12. Equalizing multicurves for partially postcritically finite maps
  13. Elements of the mapping class group associated to equalizing multicurves.
  14. Period $4$ curve
  15. Examples of of Equalizing multicurves
  16. ...and 6 more sections

Key Result

Theorem 1

$f^*\mathcal{S}_{\Gamma}=i_*\mathcal{S}_{\Gamma}$ if and only if $\Gamma$ is an equalizing multicurve. If the deformation space accumulates on $\mathcal{S}_{\Gamma}$, then there exists $\tilde{\Gamma}\subset \Gamma$ such that $\tilde{\Gamma}$ is an arithmetically equalizing multicurve.

Figures (3)

  • Figure 2: A pair of equalizing curves $\gamma$ and $\delta$. Here and later: by $x'$ we denote a point that has the same image as $x$.
  • Figure 3: A pair of equalizing curves for a map $f\in \operatorname{Per}_n$, $n$ even
  • Figure 4: A pair of equalizing curves for a map $f\in \operatorname{Per}_n$, $n$ odd

Theorems & Definitions (51)

  • Theorem
  • Theorem
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Corollary 4.3
  • Lemma 4.4
  • Lemma 4.5
  • ...and 41 more