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Abstract
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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.
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Keywords
congruence subgroup problem, automorphism groups, profinite
groups, free metabelian groups
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Mathematical Subject Classification 2010
Primary: 19B37, 20H05
Secondary: 20E18, 20E36
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Milestones
Received: 2 August 2017
Revised: 28 March 2019
Accepted: 12 April 2019
Published: 17 December 2019
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