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Abstract We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphisms. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups.
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The Borel complexity of the space of left‐orderings, low‐dimensional topology, and dynamics
Abstract We develop new tools to analyze the complexity of the conjugacy equivalence relation , whenever is a left‐orderable group. Our methods are used to demonstrate nonsmoothness of for certain groups of dynamical origin, such as certain amalgams constructed from Thompson's group . We also initiate a systematic analysis of , where is a 3‐manifold. We prove that if is not prime, then is a universal countable Borel equivalence relation, and show that in certain cases the complexity of is bounded below by the complexity of the conjugacy equivalence relation arising from the fundamental group of each of the JSJ pieces of . We also prove that if is the complement of a nontrivial knot in then is not smooth, and show how determining smoothness of for all knot manifolds is related to the L‐space conjecture.
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- Award ID(s):
- 2348819
- PAR ID:
- 10600823
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Journal of the London Mathematical Society
- Volume:
- 110
- Issue:
- 5
- ISSN:
- 0024-6107
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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