Groups with infinitely many ends acting analytically on the circle

TL;DR

This paper investigates groups acting analytically on the circle, showing that non-expanding actions with infinitely many ends are virtually free, and extends Duminy's theorem to minimal foliations with conditions on leaf ends and holonomy structures.

Contribution

It establishes that groups of analytic diffeomorphisms with infinitely many ends are virtually free under non-expanding actions and generalizes Duminy's theorem to minimal codimension one foliations.

Findings

Non-expanding actions imply the group is virtually free.

Duminy's theorem is extended to minimal foliations with conditions on leaf ends.

Holonomy pseudogroup either preserves a projective structure or leaves have infinitely many ends.

Abstract

This article takes the inspiration from two milestones in the study of non minimal actions of groups on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves and Ghys' freeness result in analytic regularity. Our first result concerns groups of analytic diffeomorphisms with infinitely many ends: if the action is non expanding, then the group is virtually free. The second result is a Duminy's theorem for minimal codimension one foliations: either non expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure.