Geodesic
Universal theories and centralizer dimensions of groups
Algebra i logika, Tome 58 (2019) no. 3, pp. 397-416.

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The exact value of the centralizer dimension is found for a free polynilpotent group and for a free group in a variety of metabelian groups of nilpotency class at most $c$. Relations between $\exists$- and $\Phi$-theories of groups are specified, in which case the concept of centralizer dimension plays an important role.
Keywords: polynilpotent group, free group, variety of metabelian groups, centralizer dimension, $\exists$-theories of groups, $\Phi$-theories of groups. 
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E. I. Timoshenko. Universal theories and centralizer dimensions of groups. Algebra i logika, Tome 58 (2019) no. 3, pp. 397-416. http://geodesic.mathdoc.fr/item/AL_2019_58_3_a7/

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