Ranks of mapping tori via the curve complex

TL;DR

This paper establishes a relationship between the translation distance of monodromy in the curve complex and the rank of the fundamental group of fibered 3-manifolds, providing a new geometric criterion for algebraic properties.

Contribution

It proves that large translation distance in the curve complex implies a specific rank for the fundamental group of fibered 3-manifolds, linking geometric and algebraic invariants.

Findings

Large translation distance implies rank 2g+1 for the fundamental group.

Connects geometric properties of monodromy to algebraic invariants of 3-manifolds.

Provides a criterion for determining fundamental group rank based on curve complex dynamics.

Abstract

We show that if the monodromy of a 3-manifold M that fibers over the circle has large translation distance in the curve complex, then the rank of the fundamental group of M is 2g+1, where g is the genus of the fiber.