 
Summary: arXiv:math.GT/0601677v127Jan2006
COVERING SPACES OF ARITHMETIC 3ORBIFOLDS
M. Lackenby, D. D. Long and A. W. Reid
1. Introduction
This paper investigates properties of finite sheeted covering spaces of arith
metic hyperbolic 3orbifolds (see §2). The main motivation is a central unresolved
question in the theory of closed hyperbolic 3manifolds; namely whether a closed
hyperbolic 3manifold is virtually Haken. Various strengthenings of this have also
been widely studied. Of specific to interest to us is the question of whether the
fundamental group of a given hyperbolic 3manifold M is large; that is to say,
some finite index subgroup of 1(M) admits a surjective homomorphism onto a
nonabelian free group. This implies that M is virtually Haken, and indeed that
M has infinite virtual first Betti number (see §2.4 for a definition). Of course,
a weaker formulation is to only ask whether the virtual first Betti number of a
closed hyperbolic 3manifold M is positive. This has been verified in many cases,
see [8] for some recent work on this. However, in general, passage from positive
virtual first Betti number to infinite virtual first Betti number is difficult, as is
passage from infinite virtual first Betti number to large. This paper makes some
progress on the latter in certain settings.
The background for our work is recent work of the first author (see for example
