Uniformly perfect finitely generated simple left orderable groups

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.