Subgroups and diversity of left-orderable small cancellation groups

Abstract

We arrange classical small cancellation constructions to produce left-orderable groups: we show that every finitely generated group is the quotient of a left-ordered small cancellation group by a finitely generated kernel (Rips construction). We give presentations of left-ordered hyperbolic small cancellation groups that are not locally indicable, and observe that the class of left-ordered small cancellation groups that are not locally indicable is quasi-isometrically diverse. Altogether, this shows that left-orderable small cancellation groups form a rich and diverse class.