The Problem of the Classification of the Nilpotent Class 2 Torsion Free Groups up to Geometric Equivalence

Abstract

In this article, we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism. This result leads to better understanding of the complexity of the problem of the classification of the quasi-varieties of the nilpotent class 2 groups. It is well known that the variety of the nilpotent class s groups is Noetherian for every s ∈ ℕ. So the problem of the classification of the quasi-varieties generated even by a single nilpotent class 2 finitely generated torsion free group is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism.

Key Words:

• Geometric equivalence
• Nilpotent groups
• Quasi-varieties

2000 Mathematics Subject Classification:

• 20F18
• 20E10

ACKNOWLEDGMENTS

This research was motivated by Prof. B. Plotkin. I would like to express my gratitude to him and to Prof. S. Margolis for the permanent attention to this work. Discussions with Prof. E. Rips, Prof. Z. Sela, Prof. E. Hrushovski, Prof. A. Mann, and Dr. E. Plotkin about the topic of this research were very useful. The talks with Prof. D. Kazhdan and Prof. Yu. Drozd directed my attention to the articles of Prof. V. Sergeichuk and his collaborators (Belitskii et al., 2005a,b; Sergeichuk, 1977, 1988). My discussions with Dr. R. Lipyanski led to a breakthrough, and I would like to express him my sincere gratitude. I also thank Dr. R. Lipyanski for his important remarks, which helped a lot in writing the article. I am thankful to all the authors of the article Belitskii et al. (2005b), which was very contributory to this research.

I also acknowledge the support of grants by the Israel Science Foundation founded by the Israeli Academy of Sciences—Center of Excellence Program.

Notes

Communicated by A. Yu. Olshanskii.