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Borel complexity for product of trees and commuting partial maps
Abstract
We prove that every acylindrical action on uniformly locally finite product of trees induces the hyperfinite orbit equivalence relation on the Roller boundary. As a byproduct, we construct an example of a standard Borel space and two commuting bounded-to-one surjective partial Borel maps that generate a universal countable Borel equivalence relation. This contrasts to Shinko-Weilacher-Yu's theorem on hyperfiniteness of bounded-to-one actions of commutative monoids.
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BibTeX
@misc{oyakawa2026borel,
title = {{Borel complexity for product of trees and commuting partial maps}},
author = {Koichi Oyakawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34068},
eprint = {2609.34068},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34068},
}
RIS
TY - UNPB
TI - Borel complexity for product of trees and commuting partial maps
AU - Koichi Oyakawa
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34068
UR - https://arxiv.org/abs/2609.34068
AB - We prove that every acylindrical action on uniformly locally finite product of trees induces the hyperfinite orbit equivalence relation on the Roller boundary. As a byproduct, we construct an example of a standard Borel space and two commuting bounded-to-one surjective partial Borel maps that generate a universal countable Borel equivalence relation. This contrasts to Shinko-Weilacher-Yu's theorem on hyperfiniteness of bounded-to-one actions of commutative monoids.
ER -