Algebraic sets in a finitely generated rigid $2$-step solvable pro-$p$-group

N. S. Romanovskii<sup>ab

a</sup> Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia
^b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

Abstract: A $2$-step solvable pro-$p$-group $G$ is said to be rigid if it contains a normal series of the form
$$ G=G_1>G_2>G_3=1 $$
such that the factor group $A=G/G_2$ is torsion-free Abelian, and the subgroup $G_2$ is also Abelian and is torsion-free as a $\mathbb Z_pA$-module, where $\mathbb Z_pA$ is the group algebra of the group $A$ over the ring of $p$-adic integers. For instance, free metabelian pro-$p$-groups of rank $\ge2$ are rigid. We give a description of algebraic sets in an arbitrary finitely generated $2$-step solvable rigid pro-$p$-group $G$, i.e., sets defined by systems of equations in one variable with coefficients in $G$.

Keywords: finitely generated $2$-step solvable rigid pro-$p$-group, algebraic set.

UDC: 512.5

Received: 02.07.2015

DOI: 10.17377/alglog.2015.54.604

 English version: Algebra and Logic, 2016, 54:6, 478–488

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