Subject: Re: Question about Betti numbers
From: Peter Linnell <linnell@calvin.math.vt.edu>
Date: Sat, 11 Feb 2006 10:33:00 -0500 (EST)

>> Subject: question
>> From: Yuli Rudyak <rudyak@math.ufl.edu>
>> Date: Tue, 7 Feb 2006 22:49:03 -0500 (EST)
>>
>> I have a question for the list.
>> Do you know an example of a finitely presented group $G$ such that 
>> $b_1(G)=0$ 
>> bit $b_2(G)>0$ (here $b_i$ is the Betti number)?

Let A be the free abelian group of rank 2 with generators a,b.
Let C denote the cyclic group of order 3 with generator x.
Make C act on A via the rule xax^{-1} = a^{-1}b, xbx^{-1} = a^{-1}.
Form the split extension G of A by C with this action, so A is a
normal subgroup of G with G/A isomorphic C.

This is a finitely presented group with first Betti number 0 and
second Betti number 1.

An example of a finitely generated torsion-free virtually abelian
group (all finitely generated virtually abelian groups are finitely
presented) with first Betti number 0 and second Betti number 1 is
given in Example 4.7 on p. 192 of
Dekimpe, Karel(B-KULK); Eick, Bettina(D-BRNS-G)
Computational aspects of group extensions and their applications in
topology. (English. English summary)
Experiment. Math. 11 (2002), no. 2, 183--200.

Peter Linnell