# Uniformly perfect finitely generated simple left orderable groups

## TL;DR

This paper proves that certain finitely generated simple left orderable groups are uniformly perfect, do not admit homogeneous quasimorphisms, and have fixed points in circle actions, revealing their complex dynamical properties.

## Contribution

The authors establish uniform perfectness of specific simple left orderable groups and analyze their dynamical actions, answering a longstanding open question.

## Key findings

- Groups are uniformly perfect, each element is a product of three commutators.
- No homogeneous quasimorphisms exist on these groups.
- Any circle action lifts to the real line and has a fixed point.

## Abstract

We show that the finitely generated simple left orderable groups $G_{\rho}$ constructed by the first two authors in arXiv:1807.06478 are uniformly perfect - each element in the group can be expressed as a product of three commutators of elements in the group. This implies that the group does not admit any homogeneous quasimorphism. Moreover, any nontrivial action of the group on the circle, which lifts to an action on the real line, admits a fixed point. Most strikingly, it follows that the groups are examples of left orderable monsters, which means that any faithful action on the real line without a global fixed point is globally contracting. This answers Question 4 from the 2018 ICM proceedings article of the third author. (This question has also been answered simultaneously and independently, using completely different methods, by Matte Bon and Triestino in arXiv:1811.12256.) To prove our results, we provide a certain characterisation of elements of the group $G_{\rho}$ which is a useful new tool in the study of these examples.