Non-left-orderable double branched coverings

TL;DR

This paper introduces the coarse presentation method to prove that the fundamental group of certain double branched coverings of links is not left-orderable, expanding known examples and providing a new analytical tool.

Contribution

It develops the coarse presentation technique allowing inequalities in group presentations to identify non-left-orderable fundamental groups of double branched coverings.

Findings

Identifies a family of links with non-left-orderable double branched coverings.

Includes many known and new examples of such links.

Provides a systematic method for proving non-left-orderability.

Abstract

We develop a method to show the fundamental group of the double branched covering of a link is not left-orderable by introducing the notion of the coarse presentation. As in the usual group presentations, a coarse presentation is given by a set of generators and relations, but inequalities are allowed as relations. By using coarse presentation, we give a family of links whose double branched covering has the non-left-orderable fundamental group. Our family of links includes many known examples and new examples.