Left-orderability of Groups Acting on Bifoliated Planes

TL;DR

This paper proves that groups acting faithfully on bifoliated planes while preserving foliation orientations are left-orderable, using end set linear orders and boundary circle identifications to establish the result.

Contribution

It introduces a novel method to establish left-orderability for groups acting on bifoliated planes, linking end structures to boundary circle actions.

Findings

Groups acting faithfully on bifoliated planes are left-orderable.

A construction of a linear order on ends of leaf spaces is developed.

An identification between ends of leaf spaces and boundary circle subsets is established.

Abstract

We prove that any group acting faithfully on a bifoliated plane while preserving the orientations of both foliations is left-orderable. The proof utilizes a construction of a linear order on the set of ends of the leaf spaces, which takes advantage of the additional structure coming with a bifoliation. Moreover, we build an identification between ends of leaf spaces of the bifoliation and subsets of the boundary circle at infinity, and use it to give a condition for the equivalence between a faithful group action on a bifoliated plane and a faithful group action on the set of ends of the leaf spaces.