Geodesic
Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination
Algebra i logika, Tome 57 (2018) no. 6, pp. 733-748.

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A group $G$ is said to be rigid if it contains a normal series $$ G=G_1>G_2>\dots>G_m>G_{m+1}=1, $$ whose quotients $G_i/G_{i+1}$ are Abelian and, treated as right $\mathbb{Z}[G/G_i]$-modules, are torsion-free. A rigid group $G$ is divisible if elements of the quotient $G_i/G_{i+1}$ are divisible by nonzero elements of the ring $\mathbb{Z}[G/G_i]$. Every rigid group is embedded in a divisible one. THEOREM. Let $G$ be a divisible rigid group. Then the coincedence of $\exists$-types of same-length tuples of elements of the group $G$ implies that these tuples are conjugate via an authomorphism of $G$. As corollaries we state that divisible rigid groups are strongly $\aleph_0$-homogeneous and that the theory of divisible $m$-rigid groups admits quantifier elimination down to a Boolean combination of $\exists$-formulas.
Mots-clés : rigid group, divisible group, quantifier elimination.
Keywords: strongly в„µ0-homogeneous group 
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     title = {Divisible {Rigid} {Groups.} {III.} {Homogeneity} and {Quantifier} {Elimination}},
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N. S. Romanovskii. Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination. Algebra i logika, Tome 57 (2018) no. 6, pp. 733-748. http://geodesic.mathdoc.fr/item/AL_2018_57_6_a5/

[1] N. S. Romanovskii, “Delimye zhestkie gruppy. Algebraicheskaya zamknutost i elementarnaya teoriya”, Algebra i logika, 56:5 (2017), 593–612 | Zbl

[2] A. G. Myasnikov, N. S. Romanovskii, “Delimye zhestkie gruppy. II. Stabilnost, nasyschennost i elementarnye podmodeli”, Algebra i logika, 57:1 (2018), 43–56 | Zbl

[3] S. Perin, R. Sklinos, “Homogenity in the free groups”, Duke Math. J., 161:13 (2012), 2635–2668 | DOI | MR | Zbl

[4] B. Poizat, A course in model theory. An introduction to contemporary mathematical logic, Universitext, Springer, New York, NY, 2000 | DOI | MR | Zbl

[5] Yu. L. Ershov, E. A. Palyutin, Matematicheskaya logika, 6-e izd., Fizmatlit, M., 2011

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

[7] N. S. Romanovskii, “Koproizvedeniya zhestkikh grupp”, Algebra i logika, 49:6 (2010), 803–818

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