EULER CHARACTERISTICS ON A CLASS OF FINITELY GENERATED NILPOTENT GROUPS

A finitely generated torsion free nilpotent group is called an F-group. To each F-group F there is associated a connected, simply connected nilpotent Lie group G(Gamma). Let TUF be the class of all 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.