Subexponential growth and $C^1$ actions on one-manifolds

TL;DR

This paper demonstrates that groups with subexponential growth can have their actions on one-manifolds smoothly realized as $C^1$ diffeomorphisms, extending the regularity of such actions.

Contribution

It establishes that groups with no finitely generated subgroup of exponential growth can realize their actions as $C^1$ diffeomorphisms on one-manifolds, a significant regularity enhancement.

Findings

Actions of such groups can be realized as $C^1$ diffeomorphisms

Every action by homeomorphisms can be semi-conjugated to a $C^1$ action

The proof uses a functional characterization of groups with subexponential growth

Abstract

Let $G$ be a countable group with no finitely generated subgroup of exponential growth. We show that every action of $G$ on a countable set preserving a linear (respectively, circular) order can be realised as the restriction of some action by $C^1$ diffeomorphisms on an interval (respectively, the circle) to an invariant subset. As a consequence, every action of $G$ by homeomorphisms on a compact connected one-manifold can be made $C^1$ upon passing to a semi-conjugate action. The proof is based on a functional characterisation of groups of local subexponential growth.