Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the asymptotic translation lengths on the sphere complexes and the generalized fibered cone

Hyungryul Baik, Dongryul M. Kim, Chenxi Wu

Abstract

In this paper, we study the asymptotic translation lengths on the sphere complexes of monodromies of a manifold fibered over the circle. Given a compact mapping torus, we define a cone in the first cohomology which we call the generalized fibered cone, and show that every primitive integral element gives a fibration over the circle. Moreover, we prove that the generalized fibered cone is a rational slice of Fried's cone, which is defined as the dual of homological directions, an analogue of Thurston's fibered cone. As a consequence of our description of the generalized fibered cone, we provide each proper subcone of the generalized fibered cone with a uniform upper bound for asymptotic translation lengths of monodromies on sphere complexes of fibers in the proper subcone. Our upper bound is purely in terms of the dimension of the proper subcone. We also deduce similar estimates for asymptotic translation lengths of some mapping classes on finite graphs constructed in the works of Dowdall--Kapovich--Leininger, measured on associated free-splitting complexes and free-factor complexes. Moreover, as an application of our result, we prove that the asymptote for the minimal asymptotic translation length of the genus $g$ handlebody group on the disk complex is $1/g^2$, the same as the one on the curve complex.

On the asymptotic translation lengths on the sphere complexes and the generalized fibered cone

Abstract

In this paper, we study the asymptotic translation lengths on the sphere complexes of monodromies of a manifold fibered over the circle. Given a compact mapping torus, we define a cone in the first cohomology which we call the generalized fibered cone, and show that every primitive integral element gives a fibration over the circle. Moreover, we prove that the generalized fibered cone is a rational slice of Fried's cone, which is defined as the dual of homological directions, an analogue of Thurston's fibered cone. As a consequence of our description of the generalized fibered cone, we provide each proper subcone of the generalized fibered cone with a uniform upper bound for asymptotic translation lengths of monodromies on sphere complexes of fibers in the proper subcone. Our upper bound is purely in terms of the dimension of the proper subcone. We also deduce similar estimates for asymptotic translation lengths of some mapping classes on finite graphs constructed in the works of Dowdall--Kapovich--Leininger, measured on associated free-splitting complexes and free-factor complexes. Moreover, as an application of our result, we prove that the asymptote for the minimal asymptotic translation length of the genus handlebody group on the disk complex is , the same as the one on the curve complex.

Paper Structure

This paper contains 14 sections, 5 theorems, 55 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalized fibered cone and asymptotic translation lengths
  3. Minimal asymptotic translation lengths on disk graphs
  4. Future directions
  5. Generalized fibered cone and slices of Fried's cone
  6. Generalized fibered cone as the slice of Fried's cone
  7. Asymptotic translation lengths on sphere complexes
  8. Minimal translation lengths of Handlebody groups
  9. Folding sequences of graph maps and (doubled) handlebodies
  10. Doubled handlebody
  11. Handlebody
  12. Free-splitting complexes and free-factor complexes
  13. Positive cone
  14. Free-splitting and free-factor complexes

Key Result

Theorem A

Let $\varphi : M \to M$ be a diffeomorphism of a compact manifold to itself. Let $N$ be the mapping torus of $\varphi$. Then the generalized fibered cone in $H^1(N)$ associated to $\varphi$ is the intersection of a rational subspace and Fried's cone for the suspension flow given by $\varphi$ in $H^1

Figures (11)

  • Figure 1: Description of $\Gamma \otimes \mathop{\mathrm{\mathbb{R}}}\nolimits$. The subspace $H \otimes \mathop{\mathrm{\mathbb{R}}}\nolimits$ is illustrated as a horizontal plane. $a_1, a_2 \in \alpha^{\perp}$ and the dotted regions are $\Omega(a_1)$ and $\Omega(a_2)$. The ball centered at $h$ is of radius $Cn_{\alpha}^{1/d}$ in $\Gamma \otimes \mathop{\mathrm{\mathbb{R}}}\nolimits$. The constant $c$ is chosen appropriately so that $h\tilde{\varphi}^{\left\lfloor cn_{\alpha}^{1/d}\right\rfloor}$ belongs to the ball.
  • Figure 2: A handlebody of genus 2
  • Figure 3: Cyclic cover $\tilde{V}$
  • Figure 4: The gluing corresponding to an edge attached to a vertex.
  • Figure 5: 3-manifolds obtained from the graphs. Light vertices in the graphs correspond to the dotted regions. Dark vertices in the graphs correspond to the outermost regions in the right figures. Edges of the graphs correspond the hatched regions diffeomorphic to $S^2 \times I$.
  • ...and 6 more figures

Theorems & Definitions (24)

  • Theorem A: Theorem \ref{['thm.genfibfried']}
  • Definition 1.1: Sphere complex
  • Remark 1.2
  • Definition 1.3: Asymptotic translation length
  • Theorem B
  • Remark 1.4
  • Definition 1.5: Minimal asymptotic translation length
  • Definition 1.6: Disk graph
  • Theorem C
  • proof
  • ...and 14 more