Foliations from left orders

TL;DR

This paper introduces a method to construct singular foliations from left orders of a manifold's fundamental group, exploring its implications especially in dimension 2 and connections to 3-dimensional topology.

Contribution

It presents a novel construction linking algebraic order structures to geometric foliations, with detailed analysis in low dimensions and applications to 3-manifold problems.

Findings

Construction produces singular foliations from left orders

Connections established between algebraic orders and geometric structures

Detailed analysis in dimension 2 with implications for 3-manifolds

Abstract

We describe a construction which takes as an input a left order of the fundamental group of a manifold, and outputs a (singular) foliation of this manifold which is analogous to a taut foliation. We investigate this construction in detail in dimension $2$, and exhibit connections to various problems in dimension $3$.