Rank gradients of infinite cyclic covers of 3-manifolds

TL;DR

This paper establishes that for 3-manifolds with no spherical boundary components, being fibered is equivalent to having zero rank and Heegaard gradients in their infinite cyclic covers, linking topological and algebraic properties.

Contribution

It proves the equivalence between fibered classes and zero rank and Heegaard gradients in the context of 3-manifolds, providing new insights into their structure.

Findings

Fibered class  zero rank gradient

Fibered class zero Heegaard gradient

Equivalence of fibered property and gradient conditions

Abstract

Given a 3-manifold M with no spherical boundary components, and a primitive class \phi in H^1(M;Z), we show that the following are equivalent: (1) \phi is a fibered class, (2) the rank gradient of (M,\phi) is zero, (3) the Heegaard gradient of (M,\phi) is zero.