TITLE: Classification of discrete subgroups of the unitary group
QUESTION [5 upvotes]: Let $U(n)$ be the unitary group. From André Weil's paper "On discrete subgroups of Lie groups" it is well known that discrete cocompact subgroups of $U(n)$ have only a finite number of generators and relations.
Does there exist a full classification of discrete subgroups of the unitary group up to isomorphism?

REPLY [4 votes]: $U(n)$ is compact. So any discrete subgroup of $U(n)$ is automatically (i) cocompact and (ii) finite. If you are interested in the classification of finite subgroups of $U(n)$, then the main result is Jordan's theorem: There is an integer $J(n)$ such that any finite subgroup of $U(n)$ has a normal abelian subgroup of index $\leq J(n)$. See Terry Tao's blog for a nice exposition. There is a small industry of improving the bounds for $J(n)$; this paper 
has some good bounds.
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