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Automorphisms of the sphere complex of an infinite graph

Thomas Hill, Michael C. Kopreski, Rebecca Rechkin, George Shaji, Brian Udall

Abstract

For a locally finite, connected graph $Γ$, let $\operatorname{Map}(Γ)$ denote the group of proper homotopy equivalences of $Γ$ up to proper homotopy. Excluding sporadic cases, we show $\operatorname{Aut}(S(M_Γ)) \cong \operatorname{Map}(Γ)$, where $\mathcal{S}(M_Γ)$ is the sphere complex of the doubled handlebody $M_Γ$ associated to $Γ$. We also construct an exhaustion of $S(M_Γ)$ by finite strongly rigid sets when $Γ$ has finite rank and finitely many rays, and an appropriate generalization otherwise.

Automorphisms of the sphere complex of an infinite graph

Abstract

For a locally finite, connected graph , let denote the group of proper homotopy equivalences of up to proper homotopy. Excluding sporadic cases, we show , where is the sphere complex of the doubled handlebody associated to . We also construct an exhaustion of by finite strongly rigid sets when has finite rank and finitely many rays, and an appropriate generalization otherwise.

Paper Structure

This paper contains 23 sections, 48 theorems, 16 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Outline of paper.
  4. Preliminaries
  5. Spheres in $3$-manifolds
  6. Intersecting spheres
  7. Full subcomplexes of $\mathcal{S}(M_\Gamma)$
  8. Pants decompositions and $M_{0,s}$
  9. Edge isomorphisms and rigidity
  10. A proof of the main theorem
  11. Dual graphs of pants decompositions
  12. Constructing a diffeomorphism
  13. Rigidity of the sphere complex for finite-type doubled handlebodies with $S^2$-boundaries
  14. Constructing a geometrically rigid set $X$.
  15. Geometric rigidity of $X$ for $M_{n,s}$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\Gamma,\Gamma'$ be two locally finite connected graphs. Suppose $f: \mathcal{S}(M_{\Gamma})\to \mathcal{S}(M_{\Gamma'})$ is an isomorphism. Then $f$ is induced by a diffeomorphism $h:M_{\Gamma} \to M_{\Gamma'}$. In particular,

Figures (16)

  • Figure 1: The construction of $M_\Gamma$ for the ladder graph. Disk cross-sections in distinct copies of $N_\Gamma$ form hemispheres of a 2-sphere in $M_\Gamma$.
  • Figure 2: The three spheres $a, a',$ and $a"$ in $M_{0,4}$ differ by a flip move. $\partial M_{0,4}$ is labeled $1$--$4$.
  • Figure 3: The $M_{1,3}$ and $M_{0,6}$ components in $M_\Gamma \setminus (\sigma \setminus \{a,b,c\})$ and $M_{\Gamma'} \setminus (f\sigma \setminus \{fa,fb,fc\})$ respectively. The figure shows one-half of the doubled handlebodies.
  • Figure 4: Various links in the proof of \ref{['lem:compatibility_sigma']}. Row 1 shows when $e_a$ and $e_{b}$ are not adjacent, Row 2 when $e_a$ and $e_{b}$ are incident on a single common vertex, and Row 3 when $b = b'$. Because $a$ and $a'$ both separate $b$ and $b'$ in $M$, the left column is identical replacing $a$ with $a'$, $\sigma$ with $\sigma'$, and $b$ with $b'$.
  • Figure 5: On the left is $M \cong M_{1,2} \setminus \partial M_{1,2}$ illustrating part (i). On the right is $M \cong M_{0,5}$ with the spheres in part (ii) drawn in.
  • ...and 11 more figures

Theorems & Definitions (104)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: ayala1990proper
  • Definition 2.2
  • Definition 2.3
  • Remark
  • Definition 2.4
  • Theorem 2.5: udall2024spherecomplexlocallyfinite
  • Lemma 2.6
  • ...and 94 more