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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000000
S 0002-9939(XX)0000-0
CONSTRUCTING NON-CONGRUENCE SUBGROUPS OF
FLEXIBLE HYPERBOLIC 3-MANIFOLD GROUPS
D. COOPER, D. D. LONG, AND M. THISTLETHWAITE
(Communicated by D. Ruberman)
Abstract. We give an explicit construction for non-congruence subgroups in
the fundamental group of a flexible hyperbolic 3-manifold.
1. Introduction
When M is a closed orientable hyperbolic 3-manifold, 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

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics