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Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig, Nansen Petrosyan, Xiaolei Wu

Abstract

We prove that the infinite family of asymptotic mapping class groups of surfaces of defined by Funar--Kapoudjian and Aramayona--Funar are of type $F_\infty$, thus answering questions of Funar-Kapoudjian-Sergiescu and Aramayona-Vlamis. As it turns out, this result is a specific instance of a much more general theorem which allows to deduce that asymptotic mapping class groups of Cantor manifolds, also introduced in this paper, are of type $F_\infty$, provide the underlying manifolds satisfy some general hypotheses. As important examples, we will obtain $F_\infty$ asymptotical mapping class groups that contain, respectively, the mapping class group of every compact surface with non-empty boundary, the automorphism group of every free group of finite rank, or infinite families of arithmetic groups. In addition, for certain types of manifolds, the homology of our asymptotic mapping class groups coincides with the stable homology of the relevant mapping class groups, as studied by Harer and Hatcher--Wahl.

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

Abstract

We prove that the infinite family of asymptotic mapping class groups of surfaces of defined by Funar--Kapoudjian and Aramayona--Funar are of type , thus answering questions of Funar-Kapoudjian-Sergiescu and Aramayona-Vlamis. As it turns out, this result is a specific instance of a much more general theorem which allows to deduce that asymptotic mapping class groups of Cantor manifolds, also introduced in this paper, are of type , provide the underlying manifolds satisfy some general hypotheses. As important examples, we will obtain asymptotical mapping class groups that contain, respectively, the mapping class group of every compact surface with non-empty boundary, the automorphism group of every free group of finite rank, or infinite families of arithmetic groups. In addition, for certain types of manifolds, the homology of our asymptotic mapping class groups coincides with the stable homology of the relevant mapping class groups, as studied by Harer and Hatcher--Wahl.

Paper Structure

This paper contains 44 sections, 126 theorems, 67 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Applications.
  3. Surfaces
  4. 3-manifolds
  5. Higher-dimensional manifolds
  6. A connection with Higman's Embedding Theorem
  7. Non-positive curvature
  8. Homology
  9. Mapping class groups
  10. Finiteness properties of mapping class groups
  11. Surfaces
  12. 3-manifolds
  13. Higher dimensions
  14. Cantor manifolds and asymptotic mapping class groups
  15. The cancellation, inclusion, and intersection properties
  16. ...and 29 more sections

Key Result

Theorem 1.1

The asymptotic mapping class groups of surfaces defined in FK04FK09AF17 are of type $F_\infty$.

Figures (7)

  • Figure 1: Construction of the Cantor manifold $C_{2,1}(O,Y)$, for $O$ a closed surface of genus 2 and $Y$ a sphere (so that $Y^2$ is a pair of pants).
  • Figure 2: Reduction, of the top paired $(3,2)$-forest diagram to the bottom one.
  • Figure 3: The difference between elementary and non-elementary simplices in the poset for $C_{2,2}(O,Y)$, where $O$ is a closed surface of genus 2 and $Y$ is a sphere. While $O_1\cup Z$ and $O_1\cup Z\cup Z"$ define an elementary $1$-simplex, the simplex defined by $O_1\cup Z$ and $O_1\cup Z\cup Z'$ is non-elementary.
  • Figure 4: A $2$-simplex in $\mathcal{P}_d(M,A)$ with $M$ a surface of genus $5$ and $A=\partial M$ with $7$ spheres.
  • Figure 5: A $1$-simplex in $\mathcal{H}(M)$ with $M$ a surface of genus $4$ with $6$ boundary spheres.
  • ...and 2 more figures

Theorems & Definitions (226)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 1.9
  • Theorem 1.10
  • Theorem 1.11
  • ...and 216 more