Abstract

Let \(\mathcal{C}\) be a root class of groups and \(\mathcal{\pi}_{1}(\mathcal{G})\) be the fundamental group of a graph \(\mathcal{G}\) of groups. We prove that if \(\mathcal{G}\) has a finite number of edges and there exists a homomorphism of \(\mathcal{\pi}_{1}(\mathcal{G})\) onto a group of \(\mathcal{C}\) acting injectively on all the edge subgroups, then \(\mathcal{\pi}_{1}(\mathcal{G})\) is residually a \(\mathcal{C}\)-group. The main result of the paper is that the inverse statement is not true for many root classes of groups. The proof of this result is based on the criterion for the fundamental group of a graph of isomorphic groups to be residually a \(\mathcal{C}\)-group, which is of independent interest.

中文翻译：

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关于群体自由建构的根类残差问题

摘要

假设\（\ mathcal {C} \）是组的根类，而\（\ mathcal {\ pi} _ {1}（\ mathcal {G}）\）是图\（\ mathcal { G} \）组。我们证明，如果\（\ mathcal {G} \）具有有限数量的边，并且存在\（\ mathcal {\ pi} _ {1}（\ mathcal {G}）\）同态到一组\（\ mathcal {C} \）对所有边缘子组进行内射，则\（\ mathcal {\ pi} _ {1}（\ mathcal {G}）\）仍然是\（\ mathcal {C} \ ）-组。本文的主要结果是，对于许多根类别的组，逆语句都不成立。该结果的证明基于同构组图的基本组残差为\（\ mathcal {C} \）-组的准则，该组是独立关注的。