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math.GR2026
A paradoxical route from hyperbolic geometry to proper proximality
D. Osin, K. Toyosawa, Z. Yang
For every integer , we introduce a new combinatorial condition PP inspired by paradoxical decompositions of groups. As increases, the property PP weakens, and…
math.OA2026
Properly proximal countable measured groupoids
Adriana Fernández Quero, Kai Toyosawa
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 equ…
math.OA2026
Relative Biexactness for Relative Hyperbolic Groups and Some Applications
Ionuţ Chifan, Kai Toyosawa, Zhiyuan Yang
In this paper, we confirm a conjecture of Ozawa and others asserting that every finitely generated, relatively hyperbolic, exact group is bi-exact (in the sense of Ozawa) relative…