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On the large scale geometry of big mapping class groups of surfaces with a unique maximal end

Rita Jiménez Rolland, Israel Morales

Abstract

Building on the work of K. Mann and K. Rafi, we analyze the large scale geometry of big mapping class groups of surfaces with a unique maximal end. We obtain a complete characterization of those that are globally CB, which does not require the tameness condition. We prove that, for surfaces with a unique maximal end, any locally CB big mapping class group is CB generated, and we give an explicit criterion for determining which big mapping class groups are CB generated. Finally, we give an example of a non-tame surface whose mapping class group is CB generated but is not globally CB.

On the large scale geometry of big mapping class groups of surfaces with a unique maximal end

Abstract

Building on the work of K. Mann and K. Rafi, we analyze the large scale geometry of big mapping class groups of surfaces with a unique maximal end. We obtain a complete characterization of those that are globally CB, which does not require the tameness condition. We prove that, for surfaces with a unique maximal end, any locally CB big mapping class group is CB generated, and we give an explicit criterion for determining which big mapping class groups are CB generated. Finally, we give an example of a non-tame surface whose mapping class group is CB generated but is not globally CB.
Paper Structure (7 sections, 22 theorems, 6 equations, 4 figures)

This paper contains 7 sections, 22 theorems, 6 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['TEO:BigMapsLocallyCB1maxEnd']}
  4. Proof of Theorems \ref{['TEO:OneMaximalEndAndLocallyCBimpliesSpaceEndsSelfSimilar']} and \ref{['TEO::ClassificationLocCBMAPsWithOneMaximalEnd']}
  5. Proof of Corollary \ref{['CORO:EndsUncountable']}
  6. Proof of Theorem \ref{['THM:CBGenerated']}
  7. Proof of Theorem \ref{['THM:Example']}

Key Result

Theorem 1.1

Let $\Sigma$ be an infinite-type surface with a unique maximal end $x$. Then $\mathrm{Map}(\Sigma)$ is locally CB if and only if there is a connected finite-type subsurface $K$ of $\Sigma$ with the following properties: Moreover, if $\mathrm{Map}(\Sigma)$ is locally CB then $\mathcal{V}_K:=\{f\in \mathrm{Map}(\Sigma):\, f\vert_K=Id_K\}$ is a CB neighborhood of the identity.

Figures (4)

  • Figure 1: a) If $\mathrm{Int}(f_U(\partial U))\subseteq U$ then $f_U(\Sigma_0)\subseteq U$. b) If $\mathrm{Ext}(f_U(\partial U))\subseteq U$ then $f_U(\Sigma \setminus U)\subseteq U$.
  • Figure 2: $\Sigma_i$ and $\Sigma_i^\prime$ are homeomorphic subsurfaces in $\Sigma$. Then there is a homeomorphism $f$ with support in $P\cup \Sigma_i \cup \Sigma_i^\prime$ that sends $\Sigma_i$ onto $\Sigma_i^\prime$.
  • Figure 3: Surface $D_{n+1}$ with one boundary component and whose space of ends has Cantor-Bendixson rank $n+1$ with $(n+1)$th-derived set homeomorphic to a Cantor set $C_{n+1}$ and, each point of $C_{n+1}$ is accumulated by points homeomorphic to $\omega^{n}+1$.
  • Figure 4: Non-tame surface with a unique maximal end and whose mapping class group is CB generated but not globally CB.

Theorems & Definitions (46)

  • Definition 1
  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Definition 2
  • Definition 3
  • Remark 2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • ...and 36 more