Shirshov的融合自由积与一般幂零群
Shirshov's amalgamated free product and generic nilpotent groups
- University of the Basque Country（巴斯克大学）
- The American University in Cairo（开罗美国大学）
- University of Notre Dame（圣母大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文采用Shirshov方法简化融合积构造，证明有限幂零群嵌入UL等价群，恢复并强化Ivanov-Majcher结果，获得共尾融合性质及Fraïssé极限的通用性。
中文摘要 AI 辅助
在早期工作中，作者基于Maier和Higman的工作中根植的复杂归纳法，给出了在固定幂零类内滤过李代数的融合自由积的构造和描述。在本文中，作者采用A. I. Shirshov关于李代数融合工作的观点和方法，给出了该融合积的新描述，大大简化了他们之前的方法。这一新描述产生了群论和描述集合论两方面的应用。一个$c$-幂零群被称为UL等价的，如果其下中心列和上中心列重合。我们证明，每个素数指数$p>c$的有限$c$-幂零群都嵌入到一个指数为$p$的有限UL等价$c$-幂零群中。这恢复了Ivanov和Majcher的一个结果，即指数$p>c$的可枚举$c$-幂零群的Polish空间具有一个余稠密轨道。我们的结果还有以下推论。首先，指数$p>c$的有限$c$-幂零群类具有共尾融合性质（回答了Ivanov和Majcher的一个问题，他们证明了该类具有弱融合性质）。其次，指数$p>c$的$c$-Lazard群的Fraïssé极限在群语言上的归约在可枚举群空间中是通用的。最后，我们还证明了这些结果对无挠$c$-幂零群的类似版本。
英文摘要
In earlier work, the authors gave a construction and description of an amalgamated free product of filtered Lie algebras within a fixed nilpotency class, based on an intricate induction rooted in the work of Maier and of Higman. In this paper, the authors give a new description of this amalgam, adopting the viewpoint and methods from the work of A. I. Shirshov on amalgamation of Lie algebras, substantially simplifying their previous approach. This new description yields both group-theoretic and descriptive set-theoretic applications. A $c$-nilpotent group is called UL-equivalent if its lower and upper central series coincide. We prove that every finite $c$-nilpotent group of prime exponent $p$ with $p>c$ embeds in a finite UL-equivalent $c$-nilpotent group of exponent $p$. This recovers a result due to Ivanov and Majcher that the Polish space of enumerated $c$-nilpotent groups of exponent $p>c$ has a comeager orbit. Our result also has the following consequences. First, the class of finite $c$-nilpotent groups of exponent $p>c$ has the \textit{cofinal} amalgamation property (answering a question of Ivanov and Majcher, who showed that it has the \textit{weak} amalgamation property). Second, the reduct of the Fraïssé limit of $c$-Lazard groups of exponent $p>c$ to the group language is generic in the space of enumerated groups. Finally, we prove analogues of these results also for torsion-free $c$-nilpotent groups.
