Ordered groups, eigenvalues, knots, surgery and L-spaces

TL;DR

This paper links the algebraic properties of automorphisms in bi-orderable groups to topological features of knots and 3-manifolds, establishing conditions under which knot groups are bi-orderable and their implications for L-spaces.

Contribution

It introduces a necessary eigenvalue condition for automorphisms of bi-orderable groups and applies this to knot theory and 3-manifold topology, connecting algebraic and geometric properties.

Findings

A bi-orderable automorphism must have a real positive eigenvalue.

Fibred knots with bi-orderable groups have Alexander polynomials with positive real roots.

Surgery on such knots cannot produce L-spaces.

Abstract

We establish a necessary condition that an automorphism of a nontrivial finitely generated bi-orderable group can preserve a bi-ordering: at least one of its eigenvalues, suitably defined, must be real and positive. Applications are given to knot theory, spaces which fibre over the circle and to the Heegaard-Floer homology of surgery manifolds. In particular, we show that if a nontrivial fibred knot has bi-orderable knot group, then its Alexander polynomial has a positive real root. This implies that many specific knot groups are not bi-orderable. We also show that if the group of a nontrivial knot is bi-orderable, surgery on the knot cannot produce an $L$-space, as defined by Ozsv\'ath and Szab\'o.