Osaka Journal of Mathematics

Euler characteristics on a class of finitely generated nilpotent groups

Hatem Hamrouni

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A finitely generated torsion free nilpotent group is called an $\mathrsfs{F}$-group. To each $\mathrsfs{F}$-group $\Gamma$ there is associated a connected, simply connected nilpotent Lie group $G_{\Gamma}$. Let TUF be the class of all $\mathrsfs{F}$-group $\Gamma$ such that $G_{\Gamma}$ is totally unimodular. A group in TUF is called TUF-group. In this paper, we are interested in finding non-zero Euler characteristic on the class TUF and therefore, on TUFF, the class of groups $K$ having a subgroup $\Gamma$ of finite index in TUF. An immediate consequence we obtain that any two isomorphic finite index subgroups of a TUFF-group have the same index. As applications, we give two results, the first is a generalization of Belegradek's result, in which we prove that every TUFF-group is co-hopfian. The second is a known result due to G.C. Smith, asserting that every TUFF-group is not compressible.

Article information

Osaka J. Math., Volume 50, Number 2 (2013), 339-346.

First available in Project Euclid: 21 June 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F18: Nilpotent groups [See also 20D15]


Hamrouni, Hatem. Euler characteristics on a class of finitely generated nilpotent groups. Osaka J. Math. 50 (2013), no. 2, 339--346.

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