保持测度的词是本原的
Measure Preserving Words are Primitive
摘要：
基于自由群中的本原元和自由因子在有限群上诱导的分布，建立了它们的新刻画。对于每个有限群$G$，在$k$生成元上的自由群中的字$w$导出从$G^k$到$G$的字映射。如果给定在$G^k$上的均匀分布，则这个词映射的图像在$G$上均匀分布，我们说$w$关于$G$是保测度的。很容易看出，原始词（属于自由群的某个基的词）是保持w.r.t.测度的。所有的有限群，并且几个作者已经证明这两个性质实际上是等价的。我们在这里证明这个猜想。证明的主要成分包括Stallings图的随机覆盖、自由群的代数扩张和莫比乌斯反演。我们的方法产生更强的结果，$F_k$的子群是测度保持的当且仅当它是一个自由因子。作为该结果的一个有趣的推论，我们解决了关于自由群的profinite拓扑的问题，并表明$F_k$的本原元形成了该拓扑中的闭集。
英文摘要：
We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. For every finite group $G$, a word $w$ in the free group on $k$ generators induces a word map from $G^k$ to $G$. We say that $w$ is measure preserving with respect to $G$ if given uniform distribution on $G^k$, the image of this word map distributes uniformly on $G$. It is easy to see that primitive words (words which belong to some basis of the free group) are measure preserving w.r.t. all finite groups, and several authors have conjectured that the two properties are, in fact, equivalent. Here we prove this conjecture. The main ingredients of the proof include random coverings of Stallings graphs, algebraic extensions of free groups, and Möbius inversions. Our methods yield the stronger result that a subgroup of $F_k$ is measure preserving if and only if it is a free factor. As an interesting corollary of this result we resolve a question on the profinite topology of free groups and show that the primitive elements of $F_k$ form a closed set in this topology.
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