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 $E_\mathsf{lo}(G)$, whenever $G$ is a left-orderable group. Our methods are used to demonstrate non-smoothness of $E_\mathsf{lo}(G)$ for certain groups $G$ of dynamical origin, such as certain amalgams constructed from Thompson's group $F$. We also initiate a systematic analysis of $E_\mathsf{lo}(\pi_1(M))$, where $M$ is a $3$-manifold. We prove that if $M$ is not prime, then $E_\mathsf{lo}(\pi_1(M))$ is a universal countable Borel equivalence relation, and show that in certain cases the complexity of $E_\mathsf{lo}(\pi_1(M))$ is bounded below by the complexity of the conjugacy equivalence relation arising from the fundamental group of each of the JSJ pieces of $M$. We also prove that if $M$ is the complement of a nontrivial knot in $S^3$ then $E_\mathsf{lo}(\pi_1(M))$ is not smooth, and show how determining smoothness of $E_\mathsf{lo}(\pi_1(M))$ for all knot manifolds $M$ is related to the L-space conjecture.