The IA-congruence kernel of high rank free Metabelian groups

Abstract

The congruence subgroup problem for a finitely generated group $\Gamma$ and $G\le Aut(\Gamma )$ asks whether the map $\hat{G}\to Aut(\hat{\Gamma })$ is injective, or more generally, what is its kernel $C\left(G,\Gamma \right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we investigate $C\left(IA({\Phi }_n),{\Phi }_n\right)$, where ${\Phi }_n$ is a free metabelian group on $n\ge 4$ generators, and $IA({\Phi }_n)=\ker (Aut({\Phi }_n)\to G{L}_n(\mathbb{Z}))$. We show that in this case $C(IA({\Phi }_n),{\Phi }_n)$ is abelian, but not trivial, and not even finitely generated. This behavior is very different from what happens for free metabelian group on $n=2,3$ generators, or for finitely generated nilpotent groups.

50 pages. arXiv admin note: substantial text overlap with arXiv:1701.02459

References in corpus (2)

• Universal lattices and Property $\tau$
• The congruence subgroup property for $Aut{F}_2$: A group-theoretic proof of Asada's theorem