 
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*
Abstract
Let d be a square free positive integer and Od the ring of integers in
Q(
d). The main result of this paper is that the groups PSL(2, Od) are
subgroup separable on geometrically finite subgroups.
1. Introduction
Let G be a group and H a finitely generated subgroup; G is called
Hsubgroup 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 Hsubgroup 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 [12] and [22] for examples.
