Abstract
We develop new tools to analyze the complexity of the conjugacy equivalence relation (Formula presented.), whenever (Formula presented.) is a left-orderable group. Our methods are used to demonstrate nonsmoothness of (Formula presented.) for certain groups (Formula presented.) of dynamical origin, such as certain amalgams constructed from Thompson's group (Formula presented.). We also initiate a systematic analysis of (Formula presented.), where (Formula presented.) is a 3-manifold. We prove that if (Formula presented.) is not prime, then (Formula presented.) is a universal countable Borel equivalence relation, and show that in certain cases the complexity of (Formula presented.) is bounded below by the complexity of the conjugacy equivalence relation arising from the fundamental group of each of the JSJ pieces of (Formula presented.). We also prove that if (Formula presented.) is the complement of a nontrivial knot in (Formula presented.) then (Formula presented.) is not smooth, and show how determining smoothness of (Formula presented.) for all knot manifolds (Formula presented.) is related to the L-space conjecture.
| Original language | American English |
|---|---|
| Article number | e70024 |
| Journal | Journal of the London Mathematical Society |
| Volume | 110 |
| Issue number | 5 |
| DOIs | |
| State | Published - Nov 2024 |
ASJC Scopus subject areas
- General Mathematics