Euler characteristics on a class of finitely generated nilpotent groups

Abstract

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.