Finitely generated infinite simple groups of homeomorphisms of the real line

TL;DR

This paper constructs finitely generated infinite simple groups of homeomorphisms of the real line, answering a long-standing open question and providing a continuum of such groups with diverse structures.

Contribution

It introduces a novel construction of finitely generated simple groups of homeomorphisms of the real line, solving Rhemtulla's open problem from 1980.

Findings

Existence of finitely generated infinite simple groups of homeomorphisms of the real line

Construction of a continuum of non-isomorphic such groups

These groups are also examples of simple left (or right) orderable groups

Abstract

We construct examples of finitely generated infinite simple groups of homeomorphisms of the real line. Equivalently, these are examples of finitely generated simple left (or right) orderable groups. This answers a well known open question of Rhemtulla from 1980 concerning the existence of such groups. In fact, our construction provides a family of continuum many isomorphism types of groups with these properties.