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?

Annalspapers by Weil (1960, 1962) involve a connected Lie group`$G$`

(usually having no compact factors) and a discrete cocompact subgroup`$\Gamma$`

("uniform lattice"). Here`$\Gamma$`

is finitely generated, as the fundamental group of a compact manifold. Classification of such discrete groups depends heavily on which Lie group one considers and is closely related to the study of compact manifolds. – Jim Humphreys Apr 14 '11 at 13:39