Summary: Annals of Mathematics, 153 (2001), 599621
The Bianchi groups are separable
on geometrically finite subgroups
By I. Agol, D. D. Long, and A. W. Reid*
Let d be a square free positive integer and Od the ring of integers in
-d). The main result of this paper is that the groups PSL(2, Od) are
subgroup separable on geometrically finite subgroups.
Let G be a group and H a finitely generated subgroup; G is called
H-subgroup separable if given any g G\H, there exists a subgroup K < G of
finite index with H < K and g / K. Also, G is called subgroup separable (or
LERF) if G is H-subgroup separable for all finitely generated H < G. Sub-
group separability is an extremely powerful property, for instance it is much
stronger than residual finiteness. The class of groups for which subgroup sepa-
rability is known for all finitely generated subgroups is extremely small: abelian
groups, free groups, surface groups and carefully controlled amalgamations of
these; see  and  for examples.