Geodesic
Geometric equivalence of groups
Trudy Instituta matematiki i mehaniki, Группы и графы, Tome 13 (2007) no. 1, pp. 57-78

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Based on the notion of geometric equivalence of groups, new classes of groups, namely, geometric varieties of groups, are defined. Some properties of such classes, including their relation to quasi-varieties and prevarieties of groups, are studied. Examples of torsion free nilpotent groups that are geometrically nonequivalent to their minimal completions, as well as an example of centrally metabelian groups that are geometrically nonequivalent but generate equal quasi-varieties, are given. 
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V. V. Bludov; B. V. Gusev. Geometric equivalence of groups. Trudy Instituta matematiki i mehaniki, Группы и графы, Tome 13 (2007) no. 1, pp. 57-78. http://geodesic.mathdoc.fr/item/TIMM_2007_13_1_a3/