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Properly proximal countable measured groupoids
Abstract
We introduce the notion of proper proximality for countable measured groupoids, extending the corresponding notion for countable groups. We prove that this groupoid property is equivalent to a strengthened form of relative proper proximality for the associated groupoid von Neumann algebra. We investigate permanence properties and show, in particular, that proper proximality is preserved under finite direct products, formation of transformation groupoids, and measure equivalence. We also identify broad classes of properly proximal groupoids, including transverse measured groupoids and free product groupoids, and prove that inner amenable groupoids are never properly proximal.
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BibTeX
@misc{quero2026properly,
title = {{Properly proximal countable measured groupoids}},
author = {Adriana Fernández Quero and Kai Toyosawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01253},
eprint = {2609.01253},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01253},
}
RIS
TY - UNPB
TI - Properly proximal countable measured groupoids
AU - Adriana Fernández Quero
AU - Kai Toyosawa
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01253
UR - https://arxiv.org/abs/2609.01253
AB - We introduce the notion of proper proximality for countable measured groupoids, extending the corresponding notion for countable groups. We prove that this groupoid property is equivalent to a strengthened form of relative proper proximality for the associated groupoid von Neumann algebra. We investigate permanence properties and show, in particular, that proper proximality is preserved under finite direct products, formation of transformation groupoids, and measure equivalence. We also identify broad classes of properly proximal groupoids, including transverse measured groupoids and free product groupoids, and prove that inner amenable groupoids are never properly proximal.
ER -