Solvable Groups and Affine Actions on the Line

TL;DR

This paper investigates the actions of finitely generated solvable groups on real intervals, establishing conditions under which these actions are semi-conjugate to affine actions, with results varying based on smoothness and group class.

Contribution

It provides new structural results on solvable group actions on intervals, including conditions for semi-conjugacy to affine actions and robustness of affine actions for certain solvable groups.

Findings

Actions by $C^1$ diffeomorphisms are semi-conjugate to affine actions.

Semi-conjugacy to affine actions exists locally near endpoints for homeomorphism actions.

Affine actions are robust for all solvable linear groups, but this does not hold for all solvable groups.

Abstract

We prove a structural result for orientation-preserving actions of finitely generated solvable groups on real intervals, considered up to semi-conjugacy. As applications we obtain new answers to a problem first considered by J. F. Plante, which asks under which conditions an action of a solvable group on a real interval is semi-conjugate to an action on the line by affine transformations. We show that this is always the case for actions by $C^1$ diffeomorphisms on closed intervals. For arbitrary actions by homeomorphisms, for which this result is no longer true (as shown by Plante), we show that a semi-conjugacy to an affine action still exists in a local sense, at the level of germs near the endpoints. Finally for a vast class of solvable groups, including all solvable linear groups, we show that the family of affine actions on the line is robust, in the sense that any action by homeomorphisms on the line which is sufficiently close to an affine action must be semi-conjugate to an affine action. This robustness fails for general solvable groups, as illustrated by a counterexample.