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arXiv · 2305.03927

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

Filippo Calderoni·Adam Clay
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Abstract

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 $E_\mathsf{lo}(π_1(M))$, where $M$ is a $3$-manifold. We prove that if $M$ is not prime, then $E_\mathsf{lo}(π_1(M))$ is a universal countable Borel equivalence relation, and show that in certain cases the complexity of $E_\mathsf{lo}(π_1(M))$ 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 $E_\mathsf{lo}(π_1(M))$ is not smooth, and show how determining smoothness of $E_\mathsf{lo}(π_1(M))$ for all knot manifolds $M$ is related to the L-space conjecture.

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BibTeXRIS

Filippo Calderoni, Adam Clay. 2024-09-30. The Borel complexity of the space of left-orderings, low-dimensional topology, and dynamics. https://arxiv.org/abs/2305.03927

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