Finitely presented simple left-orderable groups in the landscape of Richard Thompson's groups

TL;DR

This paper constructs the first finitely presented simple groups of orientation-preserving homeomorphisms of the real line, demonstrating their complex algebraic and geometric properties, including infinite geometric dimension and nontrivial quasimorphisms.

Contribution

It introduces the first examples of finitely presented simple groups acting on the real line with advanced algebraic and geometric features.

Findings

First finitely presented simple groups of homeomorphisms of the real line

Groups are of type F_infinity and have infinite geometric dimension

Existence of nontrivial homogeneous quasimorphisms

Abstract

We construct the first examples of finitely presented simple groups of orientation-preserving homeomorphisms of the real line. Our examples are also of type $F_{\infty}$, have infinite geometric dimension, and admit a nontrivial homogeneous quasimorphism (and hence have infinite commutator width).