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Let d be a square free positive integer and O d the ring of integers in Q( √ −d). The main result of this paper is to show that the groups PSL(2, O d ) are subgroup separable on geometrically finite subgroups.

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We prove that there exist finitely presented, residually finite groups that are profinitely rigid in the class of all finitely presented groups but not in the class of all finitely generated groups. These groups are of the form Γ × Γ where Γ is a profinitely rigid 3-manifold group; we describe a family of such groups with the property that if P is a finitely generated, residually finite group with P ∼ = Γ × Γ then there is an embedding P ֒→ Γ × Γ that induces the profinite isomorphism; in each case there are infinitely many non-isomorphic possibilities for P .

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Arithmetic Fuchsian and Kleinian groups can all be obtained from quaternion algebras (see [2,12]). In a series of papers ([8,9,10,11]), Takeuchi investigated and characterized arithmetic Fuchsian groups among all Fuchsian groups of finite covolume, in terms of the traces of the elements in the group. His methods are readily adaptable to Kleinian groups, and we obtain a similar characterization of arithmetic Kleinian groups in §3. Commensurability classes of Kleinian groups of finite co-volume are discussed in [2] and it is shown there that the arithmetic groups can be characterized as those having dense commensurability subgroup. Here the wide commensurability classes of arithmetic Kleinian groups are shown to be approximately in one-to-one correspondence with the isomorphism classes of the corresponding quaternion algebras (Theorem 2) and it easily follows that there are infinitely many wide commensurability classes of cocompact Kleinian groups, and hence of compact hyperbolic 3-manifolds.

Let C(Γ) be the set of isomorphism classes of the finite groups that are quotients (homomorphic images) of Γ. We investigate the extent to which C(Γ) determines Γ when Γ is a group of geometric interest. If Γ1 is a lattice in PSL(2, R) and Γ2 is a lattice in any connected Lie group, then C(Γ1) = C(Γ2) implies that Γ1 ∼ = Γ2. If F is a free group and Γ is a right-angled Artin group or a residually free group (with one extra condition), then C(F ) = C(Γ) implies that F ∼ = Γ. If Γ1 < PSL(2, C) and Γ2 < G are non-uniform arithmetic lattices, where G is a semi-simple Lie group with trivial centre and no compact factors, then C(Γ1) = C(Γ2) implies that G ∼ = PSL(2, C) and that Γ2 belongs to one of finitely many commensurability classes.These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two nonisomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them.In this section we recall some background on profinite completions and the theory of profinite groups; see [48], [50] and [52] for more details.