AI 中文总结
针对幂零群的共轭可分性增长函数，本文给出次数至多线性于幂零类、二次于Hirsch长度的显式多项式上界，并证明二步幂零情形下该界最优，同时纠正了文献错误。
AI 中文摘要
一个群被称为共轭可分的，如果每一对非共轭元素在某个有限商群中仍然非共轭。几乎多项式群，因此特别是所有有限生成的幂零群，都是共轭可分的。共轭可分性增长函数通过给出分离非共轭元素的最小有限商群 $Q$ 的阶，来衡量在有限商群中区分非共轭元素的复杂性。近期工作表明，该函数对幂零群具有多项式上界和下界，但这些估计既非显式也非最优。我们证明了改进的多项式上界，其次数至多与幂零类成线性关系，且至多与 Hirsch 长度成二次关系。对于 $2$-步幂零群，我们获得了更精确的估计，并证明这些估计在仅依赖于幂零类和 Hirsch 长度的界中是最优的。这纠正了现有文献中的一个错误。主要工具是将共轭可分性增长转化为李环，推广了之前在剩余有限情形下的工作。
英文摘要
A group is conjugacy separable if every pair of non-conjugate elements remains non-conjugate in some finite quotient. Virtually polycyclic groups, and hence in particular all finitely generated nilpotent groups, are conjugacy separable. The conjugacy separability growth function measures the complexity of distinguishing non-conjugate elements in finite quotients by giving the smallest order of a finite quotient $Q$ that separates them. Recent work showed that this function admits polynomial upper and lower bounds for nilpotent groups, but these estimates are neither explicit nor optimal. We prove improved polynomial upper bounds whose degree is at most linear in the nilpotency class and at most quadratic in the Hirsch length. For $2$-step nilpotent groups, we obtain sharper estimates and show that these estimates are optimal among bounds depending only on the nilpotency class and Hirsch length. This gives a correction of an error in the existing literature. The main tool is a translation of conjugacy separability growth into Lie rings, generalizing previous work in the case of residual finiteness.
