Smooth actions of $Aff^+ (\mathbb R)$ on compact surfaces with no fixed point: an elementary construction

TL;DR

This paper constructs a smooth, fixed point free action of the orientation-preserving affine group of the real line on any compact surface, providing an explicit elementary example.

Contribution

It offers an explicit elementary construction of smooth fixed point free actions of the affine group on compact surfaces, filling a gap in the literature.

Findings

Constructed explicit smooth fixed point free actions

Confirmed the existence of such actions on all compact surfaces

Provided an accessible method for similar constructions

Abstract

Any compact surface supports a continuous action of the orientation preserving affine group of the real line which is fixed point free (Lima and Plante). It is generally admitted that this action can be taken smooth although it is not easy finding references for it. Here one gives a such action.