Affiliation:
1. School of Mathematics and Statistics , University of Sydney , Sydney , NSW 2006 , Australia
Abstract
Abstract
We show that if a nilpotent group 𝐺 has a balanced presentation and Hirsch length
h
(
G
)
>
3
h(G)>3
, then
β
1
(
G
;
Q
)
=
2
\beta_{1}(G;\mathbb{Q})=2
.
There is one such group which is torsion-free and of Hirsch length
h
=
4
h=4
, and none with
h
=
5
h=5
.
We also construct a torsion-free nilpotent group 𝐺 with
h
(
G
)
=
6
h(G)=6
and
β
2
(
G
;
F
)
=
β
1
(
G
;
F
)
\beta_{2}(G;F)=\beta_{1}(G;F)
for all fields 𝐹.
Subject
Algebra and Number Theory
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