The congruence subgroup problem for the free metabelian group on generators
David El-Chai Ben-Ezra
The Hebrew University of Jerusalem, Israel
![The congruence subgroup problem for the free metabelian group on $n\geq4$ generators cover](/_next/image?url=https%3A%2F%2Fcontent.preview.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-ggd-volume-14-issue-3.png&w=3840&q=90)
Abstract
The congruence subgroup problem for a finitely generated group asks whether the map is injective, or more generally, what is its kernel ? Here denotes the profinite completion of . It is well known that for finitely generated free abelian groups for every , but , where is the free profinite group on countably many generators.
Considering, the free metabelian group on generators, it was also proven that and . In this paper we prove that for is abelian. So, while the dichotomy in the abelian case is between and , in the metabelian case it is between and .
Cite this article
David El-Chai Ben-Ezra, The congruence subgroup problem for the free metabelian group on generators. Groups Geom. Dyn. 14 (2020), no. 3, pp. 729–764
DOI 10.4171/GGD/561