Groups acting on the line with at most $2$ fixed points: an extension of Solodov's theorem

TL;DR

This paper extends Solodov's classical theorem by proving that groups acting on the line with elements having at most two fixed points are either abelian or semi-conjugate to an affine action, broadening the scope of the original result.

Contribution

The paper generalizes Solodov's theorem to include groups where elements can have up to two fixed points, expanding understanding of group actions on the line.

Findings

Groups with elements having at most two fixed points are either abelian or semi-conjugate to affine actions.

The extension broadens the class of group actions characterized by fixed point properties.

The result connects fixed point constraints to algebraic and dynamical properties of groups.

Abstract

A classical result by Solodov states that if a group acts on the line such that any non-trivial element has at most one fixed point, then the action is either abelian or semi-conjugate to an affine action. We show that the same holds if we relax the assumption, requiring that any non-trivial element has at most 2 fixed points.