Groups of piecewise projective homeomorphisms

TL;DR

This paper studies groups of piecewise projective homeomorphisms of the line, providing simple counterexamples to the von Neumann conjecture and enabling the establishment of various properties.

Contribution

It introduces and analyzes simple examples of such groups, demonstrating their significance in countering the von Neumann conjecture and exploring their properties.

Findings

Counterexamples to the von Neumann conjecture

Properties of piecewise projective homeomorphism groups

Simpler constructions of such groups

Abstract

The group of piecewise projective homeomorphisms of the line provides straightforward counter-examples to the so-called von Neumann conjecture. The examples are so simple that many additional properties can be established.