The congruence subgroup problem for a finitely generated group and asks whether the map is injective, or more generally, what its kernel is. Here denotes the profinite completion of . In this paper we investigate , where is a free metabelian group on generators, and .
We show that in this case is abelian, but not trivial, and not even finitely generated. This behavior is very different from what happens for a free metabelian group on or generators, or for finitely generated nilpotent groups.
"The IA-congruence kernel of high rank free metabelian groups." Ann. K-Theory 4 (3) 383 - 438, 2019. https://doi.org/10.2140/akt.2019.4.383