The Borel complexity of the space of left-orderings, low-dimensional topology, and dynamics

Filippo Calderoni, Adam Clay

Published 2023-05-06Version 1

We find new criteria to analyze the complexity of the conjugacy equivalence relation $E_\mathsf{lo}(G)$ for a given left-orderable group $G$. We show that $E_\mathsf{lo}(G)$ is universal whenever $G$ is the free product of left-orderable groups, and that $E_\mathsf{lo}(G)$ is not smooth whenever $G$ is simple and not bi-orderable, or when $G$ admits a closed $G$-invariant family of non-Conradian orderings. We apply these new criteria to study the smoothness of $E_\mathsf{lo}(G)$ when $G$ is the fundamental group of $3$-manifold, and by using tools related to the L-space conjecture we show that if $G$ is a non-cyclic knot group, then $E_\mathsf{lo}(G)$ is not smooth.