Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Congruence Subgroup Problem for finitely generated Nilpotent Groups   The congruence subgroup problem for a finitely generated group $\Gamma$ and
$G\leq Aut(\Gamma)$ 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 the case $G=Aut(\Gamma)$
we denote $C\left(\Gamma\right)=C\left(Aut(\Gamma),\Gamma\right)$.
  Let $\Gamma$ be a finitely generated group,
$\bar{\Gamma}=\Gamma/[\Gamma,\Gamma]$, and
$\Gamma^{*}=\bar{\Gamma}/tor(\bar{\Gamma})\cong\mathbb{Z}^{(d)}$. Denote
$Aut^{*}(\Gamma)=\textrm{Im}(Aut(\Gamma)\to Aut(\Gamma^{*}))\leq
GL_{d}(\mathbb{Z})$. In this paper we show that when $\Gamma$ is nilpotent,
there is a canonical isomorphism $C\left(\Gamma\right)\simeq
C(Aut^{*}(\Gamma),\Gamma^{*})$. In other words, $C\left(\Gamma\right)$ is
completely determined by the solution to the classical congruence subgroup
problem for the arithmetic group $Aut^{*}(\Gamma)$.
  In particular, in the case where $\Gamma=\Psi_{n,c}$ is a finitely generated
free nilpotent group of class $c$ on $n$ elements, we get that
$C(\Psi_{n,c})=C(\mathbb{Z}^{(n)})=\{e\}$ whenever $n\geq3$, and
$C(\Psi_{2,c})=C(\mathbb{Z}^{(2)})=\hat{F}_{\omega}$ = the free profinite group
on countable number of generators.

37
The categories of this paper are: supr-con
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
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The categories of this paper are: q-alg
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The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph
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The categories of this paper are: econ
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The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: solv-int