Geodesic
Algebraic sets in a~finitely generated rigid $2$-step solvable pro-$p$-group
Algebra i logika, Tome 54 (2015) no. 6, pp. 733-747.

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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. 
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N. S. Romanovskii. Algebraic sets in a~finitely generated rigid $2$-step solvable pro-$p$-group. Algebra i logika, Tome 54 (2015) no. 6, pp. 733-747. http://geodesic.mathdoc.fr/item/AL_2015_54_6_a3/

[1] N. S. Romanovskii, “Delimye zhëstkie gruppy”, Algebra i logika, 47:6 (2008), 762–776 | MR | Zbl

[2] N. S. Romanovskii, “Nëterovost po uravneniyam zhëstkikh razreshimykh grupp”, Algebra i logika, 48:2 (2009), 258–279 | MR | Zbl

[3] N. S. Romanovskii, “Neprivodimye algebraicheskie mnozhestva nad delimymi raspavshimisya zhëstkimi gruppami”, Algebra i logika, 48:6 (2009), 793–818 | MR | Zbl

[4] N. S. Romanovskii, “Koproizvedeniya zhëstkikh grupp”, Algebra i logika, 49:6 (2010), 803–818 | MR

[5] A. Myasnikov, N. Romanovskiy, “Krull dimension of solvable groups”, J. Algebra, 324:10 (2010), 2814–2831 | DOI | MR | Zbl

[6] A. G. Myasnikov, N. S. Romanovskii, “Ob universalnykh teoriyakh zhëstkikh razreshimykh grupp”, Algebra i logika, 50:6 (2011), 802–821 | MR | Zbl

[7] N. S. Romanovskiy, “Presentations for rigid solvable groups”, J. Group Theory, 15:6 (2012), 793–810 | DOI | MR | Zbl

[8] S. G. Afanaseva, N. S. Romanovskii, “Zhëstkie metabelevy pro-$p$-gruppy”, Algebra i logika, 53:2 (2014), 162–177 | MR | Zbl

[9] S. G. Melesheva, “Ob uravneniyakh i algebraicheskoi geometrii nad prokonechnymi gruppami”, Algebra i logika, 49:5 (2010), 654–669 | MR | Zbl

[10] G. Baumslag, A. Myasnikov, V. Remeslennikov, “Algebraic geometry over groups. I: Algebraic sets and ideal theory”, J. Algebra, 219:1 (1999), 16–79 | DOI | MR | Zbl

[11] A. Myasnikov, V. Remeslennikov, “Algebraic geometry over groups. II. Logical foundations”, J. Algebra, 234:1 (2000), 225–276 | DOI | MR | Zbl

[12] V. N. Remeslennikov, N. S. Romanovskii, “Neprivodimye algebraicheskie mnozhestva v metabelevoi gruppe”, Algebra i logika, 44:5 (2005), 601–621 | MR | Zbl

[13] N. S. Romanovskii, “Algebraicheskie mnozhestva v metabelevoi gruppe”, Algebra i logika, 46:4 (2007), 503–513 | MR | Zbl

[14] S. G. Afanaseva, “Koordinatnaya gruppa affinnogo prostranstva nad zhëstkoi metabelevoi pro-$p$-gruppoi”, Algebra i logika, 53:3 (2014), 295–299 | MR | Zbl

[15] J. S. Wilson, Profinite groups, Lond. Math. Soc. Monogr., New Ser., 19, Clarendon Press, Oxford, 1998 | Zbl

[16] V. N. Remeslennikov, “Teoremy vlozheniya dlya prokonechnykh grupp”, Izv. AN SSSR. Ser. matem., 43:2 (1979), 399–417 | MR | Zbl

[17] N. S. Romanovskii, “O vlozheniyakh Shmelkina dlya abstraktnykh i prokonechnykh grupp”, Algebra i logika, 38:5 (1999), 598–612 | MR