Two new families of finitely generated simple groups of homeomorphisms of the real line

TL;DR

This paper introduces two novel families of finitely generated simple groups acting on the real line, with unique algebraic and dynamical properties, including minimal actions on the circle and torus, expanding the landscape of known simple groups.

Contribution

It presents the first examples of finitely generated simple groups with minimal circle and torus actions, and explores their algebraic properties such as infinite commutator width and left orderability.

Findings

First finitely generated simple groups with circle minimal actions.

First finitely generated simple groups with torus minimal actions.

Groups exhibit infinite commutator width and are left orderable.

Abstract

The goal of this article is to exhibit two new families of finitely generated simple groups of homeomorphisms of $\mathbf{R}$. These families are strikingly different from existing families owing to the nature of their actions on $\mathbf{R}$, and exhibit surprising algebraic and dynamical features. In particular, one construction provides the first examples of finitely generated simple groups of homeomorphisms of the real line which also admit a minimal action by homeomorphisms on the circle. This provides new examples of finitely generated simple groups with infinite commutator width, and the first such left orderable examples. Another construction provides the first examples of finitely generated simple left orderable groups that admit minimal actions by homeomorphisms on the torus.