The Borel complexity of the space of left-orderings, low-dimensional topology, and dynamics
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912048628105216 |
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| author | Calderoni, Filippo Clay, Adam |
| author_facet | Calderoni, Filippo Clay, Adam |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03927 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Borel complexity of the space of left-orderings, low-dimensional topology, and dynamics Calderoni, Filippo Clay, Adam Logic Group Theory Geometric Topology 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. |
| title | The Borel complexity of the space of left-orderings, low-dimensional topology, and dynamics |
| topic | Logic Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2305.03927 |