Codimension one minimal foliations whose leaves have fundamental groups with the same polynomial growth

TL;DR

This paper proves that in certain minimal foliations, if each leaf's fundamental group exhibits polynomial growth of a fixed degree, then the foliation has no holonomy, revealing a link between leaf group properties and foliation dynamics.

Contribution

It establishes a new connection between polynomial growth of leaf fundamental groups and the absence of holonomy in codimension one minimal foliations.

Findings

Foliations with leaves having polynomial growth of degree k are without holonomy.

The result applies to transversely orientable, minimal foliations without vanishing cycles.

Provides conditions linking leaf fundamental group properties to foliation dynamics.

Abstract

Let $\mathcal{F}$ be a transversely orientable codimension one minimal foliation without vanishing cycles of a manifold $M$. We show that if the fundamental group of each leaf of $\mathcal{F}$ has polynomial growth of degree $k$ for some non-negative integer $k$, then the foliation $\mathcal{F}$ is without holonomy.