Left-Orderable Fundamental Groups and Dehn Surgery

Abstract

It is well known that left-orderability of a group need not be preserved under quotients. As knot groups provide a natural class of left-orderable groups, this paper studies the following question: when is left-orderability preserved under the quotient associated with Dehn surgery? We establish a condition on peripheral elements that must hold whenever a given Dehn surgery yields a manifold with left-orderable fundamental group, leading to a workable criterion used to determine when sufficiently positive Dehn surgery produces manifolds with non-left-orderable fundamental group. We apply this criterion to a range of examples, all of which are L-space knots, and demonstrate that all sufficiently large surgeries have non-left-orderable fundamental group. This behavior is analogous to the property that sufficiently positive surgery on an L-space knot always yields an L-space.