从双曲几何到恰当近端性的一条悖论路径
A paradoxical route from hyperbolic geometry to proper proximality
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中文总结 AI 辅助
本文引入组合条件PP(n)构建从负曲率到恰当近端性的插值层级，证明可数群恰当近端当且仅当满足某PP(n)，并给出非圆柱双曲群的几何刻画及无限层级实例。
中文摘要 AI 辅助
对于每个整数$n\ge 2$，我们引入一个新的组合条件PP$(n)$，其灵感来源于群的悖论分解。随着$n$增大，性质PP$(n)$减弱，由此产生的层级结构在负曲率的群论表现与恰当近端性之间进行插值。更精确地，我们证明一个可数群是恰当近端的当且仅当它满足某个$n$的PP$(n)$。另一方面，每个非圆柱双曲群满足PP$(2)$；此外，有限生成的PP$(2)$-群具有几何刻画：它们正是包含强拟凸、非循环、自由子群的群。特别地，我们得到每个可数非圆柱双曲群都是恰当近端的。最后，对于每个$n\in\mathbb N$，我们提供有限生成的恰当近端但不满足PP$(n)$的群的例子，从而证明我们的层级确实是无限的。
英文摘要
For every integer $n\ge 2$, 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 $n\in\mathbb N$, we provide examples of finitely generated properly proximal groups that do not satisfy PP$(n)$, thus proving that our hierarchy is indeed infinite.
