An orientable hyperbolic 3-manifold is isometric to the quotient of
hyperbolic 3-space H3 by a discrete torsion free subgroup G of the group Iso(H3)0 of orientation
-- preserving isometries of H3. The latter group is isomorphic to the (connected) group PGL2(C),
the real Lie group SL2(C) modulo its center $\pm 1$.
Generally, a discrete subgroup of PGL2(C) is called a Kleinian group.
The group G is said to have finite covolume if H3/G has finite volume,
and is said to be cocompact if H3/G is compact.
Among hyperbolic 3-manifolds, the ones originating with arithmetically
defined Kleinian groups form a class of special interest.
Such an arithmetically defined 3-manifold H3/G is essentially determined
(up to commensurability) by an algebraic number field k with
exactly one complex place, an arbitrary (but possibly empty) set of real places
and a quaternion algebra Dover k which ramifies (at least) at all real places of k.
These arithmetic Kleinian groups fall naturally into two classes ...