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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 commutatorsof elements in the group. This implies that the group does not admit anyhomogeneous quasimorphism. Moreover, any nontrivial action of the group on thecircle, which lifts to an action on the real line, admits a fixed point. Moststrikingly, it follows that the groups are examples of left orderable monsters,which means that any faithful action on the real line without a global fixedpoint is globally contracting. This answers Question 4 from the 2018 ICMproceedings article of the third author. (This question has also been answeredsimultaneously and independently, using completely different methods, by MatteBon and Triestino in arXiv:1811.12256.) To prove our results, we provide acertain characterisation of elements of the group $G_{\rho}$ which is a usefulnew tool in the study of these examples.

Uniformly perfect finitely generated simple left orderable groups