 
Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000000
S 00029939(XX)00000
CONSTRUCTING NONCONGRUENCE SUBGROUPS OF
FLEXIBLE HYPERBOLIC 3MANIFOLD GROUPS
D. COOPER, D. D. LONG, AND M. THISTLETHWAITE
(Communicated by D. Ruberman)
Abstract. We give an explicit construction for noncongruence subgroups in
the fundamental group of a flexible hyperbolic 3manifold.
1. Introduction
When M is a closed orientable hyperbolic 3manifold, the hyperbolic structure
gives rise to a canonical discrete faithful representation of its fundamental group
: 1(M)  SO(3, 1), unique up to conjugacy by Mostow rigidity. Denoting the
image group by , one sees easily from rigidity and finite generation that in fact
is conjugate to a subgroup of SO(3, 1; R), where R is obtained from the ring of
integers of a number field by inverting a certain finite number of elements.
This situation gives rise to a family of finite representations of the group as
follows. The ring R is Dedekind and standard results (see [12] §4.1) imply that for
any nonzero ideal I, the quotient R/I is finite. Thus the reduction homomorphisms
