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Property $P_{\text{naive}}$ for big mapping class groups

Tianyi Lou

Abstract

We study the property $P_{\text {naive }}$ of mapping class groups of surfaces of infinite type, that is, for any finite collection of non-trivial elements $h_{1},h_{2}, \cdots, h_{n}$, there exists another element $g\neq 1$ of infinite order such that for all $i$, $\langle g, h_{i}\rangle \cong \langle g \rangle * \langle h_{i} \rangle$.

Property $P_{\text{naive}}$ for big mapping class groups

Abstract

We study the property of mapping class groups of surfaces of infinite type, that is, for any finite collection of non-trivial elements , there exists another element of infinite order such that for all , .

Paper Structure

This paper contains 12 sections, 12 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Surfaces and mapping class groups
  4. Nondisplaceable subsurfaces
  5. Quasi-trees of metric spaces and WWPD
  6. Quasi-trees of curve graphs for big mapping class groups
  7. General results on the action of Map(S) on X
  8. The proof of Theorem \ref{['main']}
  9. The case of elements preserving K
  10. The case of loxodromic elements that do not preserve K
  11. The case of elliptic elements that do not preserve K
  12. The case of parabolic elements that do not preserve K

Key Result

Theorem 1.1

Let $S$ be a connected orientable surface with positive complexity, and assume that $S$ contains a nondisplaceable connected subsurface of finite type. Then $\operatorname{Map}(S)$ has the property $P_{\text{naive}}$.

Figures (1)

  • Figure 1: Examples of nondisplaceable subsurfaces.

Theorems & Definitions (23)

  • Theorem 1.1
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 13 more