Summary: Geometry & Topology 11 (2007) 22652276 2265
On the virtual Betti numbers of arithmetic hyperbolic
D D LONG
A W REID
We show that closed arithmetic hyperbolic 3manifolds with virtually positive first
Betti number have infinite virtual first Betti number. As a consequence, such mani-
folds have large fundamental group.
The main result of this paper is the following.
Theorem 1.1 Suppose that M is a closed arithmetic hyperbolic 3manifold which
virtually has positive first Betti number.
Then M has infinite virtual Betti number.
An interesting feature of our argument is that although it uses arithmetic in an essential
way, it is largely geometric; in particular there is no use of Borel's theorem . This
makes Theorem 1.1 strictly stronger than  in this setting, since no congruence
assumptions are made.
We recall that a group is said to be large if it has a subgroup of finite index which