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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 .

No abstract