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On the automorphism groups of hyperbolic manifolds

Ryan Budney, David Gabai

TL;DR

The authors show that the $(n-4)$-th homotopy group of the topological automorphism group $\mathrm{Homeo}(S^1\times D^{n-1})$ is not finitely generated for $n\ge 4$, and derive that the smooth and topological automorphism groups of complete finite-volume hyperbolic $n$-manifolds ($n\ge 4$) do not have the homotopy-type of finite CW-complexes. Central to the argument are barbell diffeomorphisms and the derived $\delta_k$ families, whose $W_3$ and $W'_3$ invariants yield infinite families of independent elements in $\pi_{n-4}$ of the diffeomorphism and homeomorphism groups. Section 4 extends the obstruction to the topological category using orbit configuration spaces and a topological Taylor-tower viewpoint, while Section 5 applies the construction to hyperbolic manifolds, implanting $\delta_k$ along closed geodesics and lifting to covers to produce infinite-rank abelian subgroups of $\pi_0\mathrm{Diff}_0(N)$, with a generalization to $n>4$ via summing over geodesics. Overall, the work significantly lowers the dimension threshold for non-finite-type automorphism groups in hyperbolic geometry and provides explicit geometric and homotopical constructions to detect infinite algebraic structure in these groups.

Abstract

Let Diff(N) and Homeo(N) denote the smooth and topological group of automorphisms respectively that fix the boundary of the n-manifold N, pointwise. We show that the (n-4)-th homotopy group of Homeo(S^1 \times D^{n-1}) is not finitely-generated for n >= 4 and in particular the topological mapping-class group of S^1\times D^3 is infinitely generated. We apply this to show that the smooth and topological automorphism groups of finite-volume hyperbolic n-manifolds (when n >= 4) do not have the homotopy-type of finite CW-complexes, results previously known for n >= 11 by Farrell and Jones. In particular, we show that if N is a closed hyperbolic n-manifold, and if Diff_0(N) represents the subgroup of diffeomorphisms that are homotopic to the identity, then the (n-4)-th homotopy group of Diff_0(N) is infinitely generated and hence if n=4, then π_0\Diff_0(N) is infinitely generated with similar results holding topologically.

On the automorphism groups of hyperbolic manifolds

TL;DR

The authors show that the -th homotopy group of the topological automorphism group is not finitely generated for , and derive that the smooth and topological automorphism groups of complete finite-volume hyperbolic -manifolds () do not have the homotopy-type of finite CW-complexes. Central to the argument are barbell diffeomorphisms and the derived families, whose and invariants yield infinite families of independent elements in of the diffeomorphism and homeomorphism groups. Section 4 extends the obstruction to the topological category using orbit configuration spaces and a topological Taylor-tower viewpoint, while Section 5 applies the construction to hyperbolic manifolds, implanting along closed geodesics and lifting to covers to produce infinite-rank abelian subgroups of , with a generalization to via summing over geodesics. Overall, the work significantly lowers the dimension threshold for non-finite-type automorphism groups in hyperbolic geometry and provides explicit geometric and homotopical constructions to detect infinite algebraic structure in these groups.

Abstract

Let Diff(N) and Homeo(N) denote the smooth and topological group of automorphisms respectively that fix the boundary of the n-manifold N, pointwise. We show that the (n-4)-th homotopy group of Homeo(S^1 \times D^{n-1}) is not finitely-generated for n >= 4 and in particular the topological mapping-class group of S^1\times D^3 is infinitely generated. We apply this to show that the smooth and topological automorphism groups of finite-volume hyperbolic n-manifolds (when n >= 4) do not have the homotopy-type of finite CW-complexes, results previously known for n >= 11 by Farrell and Jones. In particular, we show that if N is a closed hyperbolic n-manifold, and if Diff_0(N) represents the subgroup of diffeomorphisms that are homotopic to the identity, then the (n-4)-th homotopy group of Diff_0(N) is infinitely generated and hence if n=4, then π_0\Diff_0(N) is infinitely generated with similar results holding topologically.
Paper Structure (5 sections, 17 theorems, 64 equations, 17 figures)

This paper contains 5 sections, 17 theorems, 64 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Barbell diffeomorphisms
  3. $\pi_{n-4} {\mathrm{Diff}}(S^1 \times D^{n-1})$ and the $\delta_k$ diffeomorphisms.
  4. Homeomorphisms of $S^1 \times D^{n-1}$
  5. Automorphisms of complete finite volume hyperbolic manifolds

Key Result

Theorem 1.1

$\pi_{n-4} {\mathrm{Homeo}}(S^1 \times D^{n-1})$ is infinitely generated and in particular $\pi_0{\mathrm{Homeo}}(S^1\times D^3)$ is infinitely generated.

Figures (17)

  • Figure 1: Barbell diffeomorphism via resolution of double point.
  • Figure 2: Barbell diffeomorphism family restricted to mid-ball as map $D^{2n-6} \to {\mathrm{Emb}}(I, \mathcal{B}_{n-2,n-2}^n)$.
  • Figure 3: The composite $\Omega^{n-j} S^i \to {\mathrm{Diff}}(\mathcal{B}_{i,j}) \to \Omega^{n-i} S^j$ in a fiber over a point in $D^{j-1}$. The image of $\{*\} \times D^{j+1-i}$ from the $S^i \times D^{j+1-i}$ summand is in blue, and the image of $\{*\} \times D^1$ from the $S^j \times D^1$ summand is in red.
  • Figure 4: Null-pseudoisotopy via embedded null isotopy. Embedded handle in red.
  • Figure 5: Surgery description of a Dehn twist
  • ...and 12 more figures

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 18 more