Growth of some transversely homogeneous foliations

TL;DR

This paper investigates the growth behavior of leaves in transversely homogeneous foliations on compact manifolds, showing a dichotomy between polynomial and exponential growth based on the holonomy group's properties.

Contribution

It extends previous results by Breuillard and Gelander, providing a clear dichotomy for leaf growth in foliations with connected holonomy closure.

Findings

Leaves have either polynomial or exponential growth.

The growth type is uniform across all leaves under the given conditions.

Examples of such foliations include transversely projective cases.

Abstract

For transversely homogeneous foliations on compact manifolds whose global holonomy group has connected closure, it is shown that either all holonomy covers of the leaves have polynomial growth with degree bounded by a common constant, or all holonomy covers of the leaves have exponential growth. This is an extension of a recent answer given by Breuillard and Gelander to a question of Carri\`ere. Examples of transversely projective foliations satisfying the above condition were constructed by Chihi and ben Ramdane.