发表机构
Faculty of Mechanics and Mathematics of Moscow State University; Department of Mathematics, Vanderbilt University（莫斯科国立大学力学与数学系; 范德堡大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明有限生成可解群为幂零群当且仅当含任意 unimodular 方程解，类似结论适用于李代数，还构造了解非唯一的 unimodular 方程例子，且幂零群上的 unimodular 映射集合在复合下构成群。
AI 中文摘要
我们的结果特别表明，有限生成可解群 $G$ 是幂零群当且仅当它包含任意 unimodular 方程的解，该方程形式为 $\u220F g_i x^{n_i}=1$，其中 $g_i\in G$ 且 $\u2211 n_i=\pm1$。类似结论对李代数也成立。我们还构造了一个有限生成群 $G$ 上的 unimodular 方程 $w(x)=g$，它对任意 $g\in G$ 都有解，但部分 $g\in G$ 的解不唯一。我们证明，对幂零群 $G$，自然定义的 unimodular 映射集合 $G^n\to G^n$ 在复合运算下构成一个群。
英文摘要
Our results implies, in particular, that a finitely generated solvable group $G$ is nilpotent if and only if it contains a solution to any unimodular equation, i.e., an equation of the form $\prod g_ix^{n_i}=1$, where $g_i\in G$ and $\sum n_i=\pm1$. A similar fact turns out to be true for Lie algebras. We also exhibit an example of a unimodular equation $w(x)=g$ over a finitely generated group $G$, which has a solution (in $G$) for any $g\in G$, but the solution is not unique for some $g\in G$. We show that, for nilpotent groups $G$, the set of unimodular mappings $G^n\to G^n$ (which are defined naturally) forms a group under the composition.
Comments9 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V2: An open question is replaced with an answer due to Denis Osin
