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

TL;DR

This paper investigates the asymptotic translation lengths of monodromies on sphere complexes in fibered manifolds, introducing the generalized fibered cone and providing uniform bounds and applications to handlebody groups.

Contribution

It defines the generalized fibered cone, relates it to Fried's cone, and establishes uniform bounds on translation lengths for monodromies in this cone, with applications to handlebody groups.

Findings

The generalized fibered cone is a rational slice of Fried's cone.

Uniform upper bounds for translation lengths depend only on the subcone dimension.

Asymptotic translation length for genus g handlebody group is proportional to 1/g^2.

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.