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

We develop new tools to analyze the complexity of the conjugacy equivalence relation Elo(G)$E_\mathsf {lo}(G)$, whenever G$G$ is a left-orderable group. Our methods are used to demonstrate nonsmoothness of Elo(G)$E_\mathsf {lo}(G)$ for certain groups G$G$ of dynamical origin, such as certain amalgams constructed from Thompson's group F$F$. We also initiate a systematic analysis of Elo(pi 1(M))$E_\mathsf {lo}(\pi _1(M))$, where M$M$ is a 3-manifold. We prove that if M$M$ is not prime, then Elo(pi 1(M))$E_\mathsf {lo}(\pi _1(M))$ is a universal countable Borel equivalence relation, and show that in certain cases the complexity of Elo(pi 1(M))$E_\mathsf {lo}(\pi _1(M))$ is bounded below by the complexity of the conjugacy equivalence relation arising from the fundamental group of each of the JSJ pieces of M$M$. We also prove that if M$M$ is the complement of a nontrivial knot in S3$S<^>3$ then Elo(pi 1(M))$E_\mathsf {lo}(\pi _1(M))$ is not smooth, and show how determining smoothness of Elo(pi 1(M))$E_\mathsf {lo}(\pi _1(M))$ for all knot manifolds M$M$ is related to the L-space conjecture.