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Geometry & Topology 11 (2007) 22652276 2265 On the virtual Betti numbers of arithmetic hyperbolic
 

Summary: Geometry & Topology 11 (2007) 2265­2276 2265
On the virtual Betti numbers of arithmetic hyperbolic
3­manifolds
D COOPER
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.
57M10
1 Introduction
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 [2]. This
makes Theorem 1.1 strictly stronger than [2] 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

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara
Reid, Alan - Department of Mathematics, University of Texas at Austin

 

Collections: Mathematics