Geodesic
On some properties of free Abelian extensions
Sbornik. Mathematics, Tome 67 (1990) no. 1, pp. 303-315 
Yu. V. Kuz'min. On some properties of free Abelian extensions. Sbornik. Mathematics, Tome 67 (1990) no. 1, pp. 303-315. http://geodesic.mathdoc.fr/item/SM_1990_67_1_a17/
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     author = {Yu. V. Kuz'min},
     title = {On some properties of free {Abelian} extensions},
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     volume = {67},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1990_67_1_a17/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $B$ be an arbitrary group. Present it as a factor group of a free group $F$: $B\cong F/N$, $N\vartriangleleft F$. The extension $$1\to N/N'\to F/N'\to B\to1$$ is said to be a it free Abelian extension of the group $B$ (it is free in the category of extensions of $B$ by all possible Abelian groups). The author continues his study of the integral homology groups $H_n(F/N')$. The main result is that for any $B$ the exponent of the torsion subgroup of the group $H_n(F/N')\otimes Z[1/2]$ divides $n$ (as usual, $Z[1/2]$ is the ring of $2$-rational numbers). At the end of the paper the author formulates a number of conjectures on the homology of groups of the form $F/N'$. The notion of homological identity of a group is introduced, and the problem of describing the homological identities of free solvable and free nilpotent groups is posed. Bibliography: 9 titles.

[1] Pchelintsev S. V., “O nilpotentnykh elementakh i nilradikalakh alternykh algebr”, Algebra i logika, 24:6 (1983), 674–695 | MR

[2] Kuzmin Yu. V., “Gomologii grupp vida $F/N'$”, DAH SSSR, 296:12 (1987), 267–270

[3] Kuzmin Yu. V., “Stroenie svobodnykh grupp nekotorykh mnogoobrazii”, Matem. sb., 125(167) (1984), 128–142 | MR | Zbl

[4] Kuzmin Yu. V., “Ob elementakh konechnogo poryadka v svobodnykh gruppakh nekotorykh mnogoobrazii”, Matem. sb., 119(161) (1982), 119–131 | MR | Zbl

[5] Lindon R., Shupp P., Kombinatornaya teoriya grupp, Mir, M., 1980 | MR

[6] Gupta C. K., “The free centre-by-metabelian groups”, J. Austral. Math. Soc., 16 (1973), 294–299 | DOI | MR | Zbl

[7] Kuzmin Yu. V., “Svobodnye tsentralno-metabelevy gruppy, algebry Li i $\mathscr D$-gruppy”, Izv. AN SSSR. Ser. matem., 41:1 (1977), 3–33 | MR | Zbl

[8] Kuzmin Yu. V., “Elementy poryadka $2$ v svobodnoi tsentralno-razreshimoi gruppe”, Matem. zametki, 37:5 (1985), 643–652 | MR | Zbl