A paradoxical route from hyperbolic geometry to proper proximality

@misc{osin2026paradoxical,
  title = {{A paradoxical route from hyperbolic geometry to proper proximality}},
  author = {D. Osin and K. Toyosawa and Z. Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21896},
  eprint = {2609.21896},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21896},
}

TY  - UNPB
TI  - A paradoxical route from hyperbolic geometry to proper proximality
AU  - D. Osin
AU  - K. Toyosawa
AU  - Z. Yang
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21896
UR  - https://arxiv.org/abs/2609.21896
AB  - For every integer n≥ 2, we introduce a new combinatorial condition PP(n) inspired by paradoxical decompositions of groups. As n increases, the property PP(n) weakens, and the resulting hierarchy interpolates between group-theoretic manifestations of negative curvature and proper proximality. More precisely, we prove that a countable group is properly proximal if and only if it satisfies PP(n) for some n. On the other hand, every acylindrically hyperbolic group satisfies PP(2); furthermore, finitely generated PP(2)-groups admit a geometric characterization: they are precisely the groups containing strongly quasi-convex, non-cyclic, free subgroups. In particular, we obtain that every countable acylindrically hyperbolic group is properly proximal. Finally, for every n∈ℕ, we provide examples of finitely generated properly proximal groups that do not satisfy PP(n), thus proving that our hierarchy is indeed infinite.
ER  -