Finitely presented left orderable monsters

TL;DR

This paper constructs the first examples of finitely presented left orderable groups with complex dynamical properties, answering a longstanding question and expanding the known landscape of such groups.

Contribution

It provides the first finitely presentable and type $F_$ examples of left orderable monsters, with additional properties like non-simplicity and minimal circle actions.

Findings

First finitely presented left orderable monsters constructed.

Groups are not simple and act minimally on the circle.

Existence of infinitely many isomorphism classes of such groups.

Abstract

A left orderable monster is a finitely generated left orderable group all of whose fixpoint-free actions on the line are proximal: the action is semiconjugate to a minimal action so that for every bounded interval $I$ and open interval $J$, there is a group element that sends $I$ into $J$. In his 2018 ICM address, Navas asked about the existence of left orderable monsters. By now there are several examples, all of which are finitely generated but not finitely presentable. We provide the first examples of left orderable monsters that are finitely presentable, and even of type $F_\infty$. The construction itself is elementary, and these groups satisfy several additional properties separating them from the previous examples: they are not simple, they act minimally on the circle, and they have an infinite-dimensional space of homogeneous quasimorphisms. Our construction is flexible enough that it produces infinitely many isomorphism classes of finitely presented (and type $F_{\infty}$) left orderable monsters.