There exist finitely generated infinite simple left orderable groups

You need to know: Group, finite and infinite group, simple group, total order on a set.

Background: A generating set of a group G is a subset such that every can be written as a finite product of elements of S and their inverses. Group G is called finitely generated if it has a finite generating set S. A group G with group operation + is called left orderable if there exists a total order on G such that implies for all .

The Theorem: On 17th July 2018, James Hyde and Yash Lodha submitted to arxiv a paper in which they proved the existence of finitely generated infinite simple left orderable groups.

Short context: The Theorem answered a long-standing open question posed by Rhemtulla in 1980. In fact, the authors proved the existence of continuum many isomorphism types of such groups. The same is true for right orderable groups (groups G with a total order such that implies for all ).

Links: Free arxiv version of the original paper is here, journal version is here.

Go to the list of all theorems