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arXiv:math.GT/0601677v127Jan2006 COVERING SPACES OF ARITHMETIC 3-ORBIFOLDS
 

Summary: arXiv:math.GT/0601677v127Jan2006
COVERING SPACES OF ARITHMETIC 3-ORBIFOLDS
M. Lackenby, D. D. Long and A. W. Reid
1. Introduction
This paper investigates properties of finite sheeted covering spaces of arith-
metic hyperbolic 3-orbifolds (see 2). The main motivation is a central unresolved
question in the theory of closed hyperbolic 3-manifolds; namely whether a closed
hyperbolic 3-manifold 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 3-manifold M is large; that is to say,
some finite index subgroup of 1(M) admits a surjective homomorphism onto a
non-abelian 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 3-manifold 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

  

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

 

Collections: Mathematics