nt.number theory - cocompact discrete subgroups of SL_2

For the arithmetic point of view you mention, Maclachlan and Reid's book "The arithmetic of hyperbolic 3-manifolds" is a great reference.

In case you're interested, there are also many geometric ways of doing this too (though you might object that some of them don't explicitly give you the subgroup).

You can explicitly build closed hyperbolic 3-manifolds out of polyhedra.

You can take a finite covolume subgroup and produce a cocompact subgroup via hyperbolic dehn surgery.

By a theorem of R. Brooks ("Circle packings and co-compact extensions of Kleinian groups", Invent. math. 86, 461-469 (1986)), you can take any cocompact subgroup of $mathrm{SL}_2(mathbb{R})$, and after a small quasiconformal conjugacy, it will lie in a cocompact subgroup of $mathrm{SL}_2(mathbb{C})$.