Geodesic
Partially commutative metabelian groups: centralizers and elementary equivalence
Algebra i logika, Tome 48 (2009) no. 3, pp. 309-341.

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For partially commutative metabelian groups, annihilators of elements of commutator subgroups are described; canonical representations of elements are defined; approximability by torsion-free nilpotent groups is proved; centralizers of elements are described. Also, it is proved that two partially commutative metabelian groups have equal elementary theories iff their defining graphs are isomorphic, and that every partially commutative metabelian group is embeddable in a metabelian group with decidable universal theory.
Keywords: metabelian group, centralizer, annihilator, elementary equivalence. 
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Ch. K. Gupta; E. I. Timoshenko. Partially commutative metabelian groups: centralizers and elementary equivalence. Algebra i logika, Tome 48 (2009) no. 3, pp. 309-341. http://geodesic.mathdoc.fr/item/AL_2009_48_3_a1/

[1] T. Hsu, D. Wise, “On linear and residual properties of graph products”, Mich. Math. J., 46:2 (1999), 251–259 | DOI | MR | Zbl

[2] J. Crisp, B. Wiest, “Embeddings of graph braid groups and surface groups in right-angled Artin groups and braid groups”, Algebr., Geom., Topol., 4 (2004), 439–472 | DOI | MR | Zbl

[3] C. Droms, “Isomorphisms of graph groups”, Proc. Am. Math. Soc., 100 (1987), 407–408 | DOI | MR | Zbl

[4] K. Hang Kim, L. Makar-Limanov, J. Neggers, F. W. Roush, “Graph Algebras”, J. Algebra, 64 (1980), 46–51 | DOI | MR | Zbl

[5] M. R. Laurence, “A generating set for the automorphism group of a graph group”, J. Lond. Math. Soc. II Ser., 52:2 (1995), 318–334 | MR | Zbl

[6] A. Duncan, I. V. Kazachkov, V. N. Remeslennikov, Automorphisms of partially commutative groups. I: linear subgroups, , 14 Mar. 2008, 25 pp. arxiv: 0803.2213v1[math.GR]

[7] G. Duchamp, D. Krob, “Partially commutative Magnus transformations”, Int. J. Algebra Comput., 3:1 (1993), 15–41 | DOI | MR | Zbl

[8] E. E. Esyp, I. V. Kazachkov, V. N. Remeslennikov, “Divisibility theory and complexity of algorithms for free partially commutative groups”, Groups, languages, algorithms, Proc. the AMS-ASL joint special sess. interactions between logic, group theory, and comput. sci. (Baltimore, MD, USA, January 16–19, 2003), Contemp. Math., 378, ed. A. V. Borovik, Am. Math. Soc., Providence, RI, 2005, 319–348 | MR | Zbl

[9] C. Wrathall, “Free partially commutative groups”, Combinatorics, computing, complexity, Pap. Int. Symp. (Tianjing and Beijing/China, 1988), Math. Appl., Chin. Ser., 1, 1989, 195–216 | MR | Zbl

[10] A. J. Duncan, I. V. Kazachkov, V. N. Remeslennikov, “Parabolic and quasiparabolic subgroups of free partially commutative groups”, J. Algebra, 318:2 (2007), 918–932 | DOI | MR | Zbl

[11] A. J. Duncan, I. V. Kazachkov, V. N. Remeslennikov, “Centraliser dimension of partially commutative groups”, Geom. Dedicata, 120 (2006), 73–97 | DOI | MR | Zbl

[12] A. J. Duncan, I. V. Kazachkov, V. N. Remeslennikov, “Centraliser dimension and universal classes of groups”, Sib. elektron. matem. izv., 3 (2006), 197–215 | MR | Zbl

[13] M. Casals-Ruiz, I. V. Kazachkov, Elements of algebraic geometry and the positive theory of partially commutative groups, , 22 Oct. 2007 arxiv: 0710.4077v1[math.GR]

[14] A. J. Duncan, I. V. Kazachkov, V. N. Remeslennikov, “Orthogonal systems in finite graphs”, Sib. elektron. matem. izv., 5 (2008), 151–176

[15] A. A. Mischenko, A. V. Treier, “Struktura tsentralizatorov dlya chastichno kommutativnoi dvustupenno nilpotentnoi $Q$-gruppy”, Vest. Omsk. gos. un-ta, Spets. vypusk “Kombinatornye metody algebry i slozhnost vychislenii” (2007), 98–102

[16] A. A. Mischenko, A. V. Treier, “Grafy kommutativnosti dlya chastichno kommutativnykh dvustupenno nilpotentnykh $Q$-grupp”, Sib. elektron. matem. izv., 4 (2007), 460–481 | MR

[17] V. N. Remeslennikov, V. G. Sokolov, “Nekotorye svoistva vlozheniya Magnusa”, Algebra i logika, 9:5 (1970), 566–578 | MR | Zbl

[18] R. M. Bryant, V. A. Roman'kov, “Automorphisms groups of relatively free groups”, Math. Proc. Camb. Philos. Soc., 127:3 (1999), 411–424 | DOI | MR | Zbl

[19] N. S. Romanovskii, “K teoreme o svobode dlya proizvedenii grupp”, Algebra i logika, 38:3 (1999), 354–367 | MR

[20] Yu. V. Kuzmin, “Approksimatsiya metabelevykh grupp”, Algebra i logika, 13:3 (1974), 300–310 | MR

[21] V. N. Remeslennikov, R. Stor, On the quasivariety generated by non-ciclic free metabelian group, Preprint No 2001.20, Manchester Centre Pure Math., Manchester, 2001

[22] E. I. Timoshenko, “Ob universalnykh teoriyakh metabelevykh grupp i vlozhenii Shmelkina”, Sib. matem. zh., 42:5 (2001), 1168–1175 | MR | Zbl

[23] O. Chapuis, “Universal theory of certain solvable groups and bounded Ore group rings”, J. Algebra, 176:2 (1995), 368–391 | DOI | MR | Zbl