## Osaka Journal of Mathematics

### Euler characteristics on a class of finitely generated nilpotent groups

Hatem Hamrouni

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 21 June 2013

https://projecteuclid.org/euclid.ojm/1371833488

Mathematical Reviews number (MathSciNet)
MR3080803

Zentralblatt MATH identifier
1287.20048

Subjects

#### Citation

Hamrouni, Hatem. Euler characteristics on a class of finitely generated nilpotent groups. Osaka J. Math. 50 (2013), no. 2, 339--346. https://projecteuclid.org/euclid.ojm/1371833488

#### References

• I. Belegradek: On co-Hopfian nilpotent groups, Bull. London Math. Soc. 35 (2003), 805–811.
• I.M. Chiswell: Euler characteristics of groups, Math. Z. 147 (1976), 1–11.
• L.J. Corwin and F.P. Greenleaf: Representations of Nilpotent Lie Groups and Their Applications, Part I, Cambridge Studies in Advanced Mathematics 18, Cambridge Univ. Press, Cambridge, 1990.
• J. Dixmier and W.G. Lister: Derivations of nilpotent Lie algebras, Proc. Amer. Math. Soc. 8 (1957), 155–158.
• A. Karrass, A. Pietrowski and D. Solitar: An improved subgroup theorem for HNN groups with some applications, Canad. J. Math. 26 (1974), 214–224.
• G. Leger and S. Tôgô: Characteristically nilpotent Lie algebras, Duke Math. J. 26 (1959), 623–628.
• \begingroup A.M. Macbeath and S. Świerczkowski: Limits of lattices in a compactly generated group, Canad. J. Math. 12 (1960), 427–437. \endgroup
• A.I. Malcev: On a class of homogeneous spaces, Amer. Math. Soc. Translation 1951 (1951).
• Y. Matsushima: On the discrete subgroups and homogeneous spaces of nilpotent Lie groups, Nagoya Math. J. 2 (1951), 95–110.
• L.P. Polek: Small neighborhoods of the identity of a real nilpotent group, Proc. Amer. Math. Soc. 42 (1973), 627–630.
• M.S. Raghunathan: Discrete Subgroups of Lie Groups, Springer, New York, 1972.
• A. Reznikov: Volumes of discrete groups and topological complexity of homology spheres, Math. Ann. 306 (1996), 547–554.
• G.C. Smith: Compressibility in nilpotent groups, Bull. London Math. Soc. 17 (1985), 453–457.
• S. Yamaguchi: Derivations and affine structures of some nilpotent Lie algebras, Mem. Fac. Sci. Kyushu Univ. Ser. A 34 (1980), 151–170.
• C.T.C. Wall: Rational Euler characteristics, Proc. Cambridge Philos. Soc. 57 (1961), 182–184.