Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-amenability of mapping class groups of infinite-type surfaces and graphs

Yusen Long

Abstract

This paper completely determines the non-amenability of the mapping class groups of infinite-type surfaces, the mapping class groups of locally finite infinite graphs of higher ranks, gives an example of non-amenable stabiliser of a point at infinity of a coarsely bounded generated hyperbolic Polish group, and exhibits a class of mapping class groups of trees or rank-one graphs that are amenable.

Non-amenability of mapping class groups of infinite-type surfaces and graphs

Abstract

This paper completely determines the non-amenability of the mapping class groups of infinite-type surfaces, the mapping class groups of locally finite infinite graphs of higher ranks, gives an example of non-amenable stabiliser of a point at infinity of a coarsely bounded generated hyperbolic Polish group, and exhibits a class of mapping class groups of trees or rank-one graphs that are amenable.

Paper Structure

This paper contains 8 sections, 18 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Amenability.
  4. Mapping class group of surfaces.
  5. Mapping class group of graphs.
  6. Mapping class groups of infinite-type surfaces
  7. Mapping class groups of infinite-type graphs
  8. Mapping class groups of trees

Key Result

Theorem 1.1

Let $S$ be an infinite-type surface. Then every open subgroup of $\mathop{\mathrm{\mathcal{MCG}}}\nolimits(S)$ is not amenable. In particular, the group $\mathop{\mathrm{\mathcal{MCG}}}\nolimits(S)$ itself is not amenable.

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Proposition 2.1: Proposition 4.1, grigorchuk2017amenability
  • Lemma 3.1
  • proof
  • ...and 23 more