Geodesic
On the growth of rank for subgroups of finitely generated groups
Sbornik. Mathematics, Tome 190 (1999) no. 8, pp. 1151-1172 
D. V. Osin. On the growth of rank for subgroups of finitely generated groups. Sbornik. Mathematics, Tome 190 (1999) no. 8, pp. 1151-1172. http://geodesic.mathdoc.fr/item/SM_1999_190_8_a2/
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     title = {On the growth of rank for subgroups of finitely generated groups},
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Voir la notice de l'article provenant de la source Math-Net.Ru

In [1] and [2] the functions of rank growth were independently introduced and investigated for subgroups of a finitely generated free group. In the present paper the concept of growth of rank is extended to subgroups of an arbitrary finitely generated group $G$, and the dependence of the asymptotic behaviour of the above functions on the choice of a finite generating set in $G$ is studied. For a broad class of groups (which includes, in particular, the free polynilpotent groups) estimates for the growth of rank for subgroups are obtained that generalize the wellknown Baumslag–Eidel'kind result on finitely generated normal subgroups. Some problems related to the realization of arbitrary functions as functions of rank growth for subgroups of soluble groups are treated.

[1] Osin D. V., “On the generating set's growth functions for subgroups of free groups”, Internat. J. Algebra Comput., 9:1 (1999), 41–50 | DOI | MR | Zbl

[2] Rosenmann A., When Schreier transversals grow wild, Preprint, 1997 | MR | Zbl

[3] Howson A. G., “On the intersection of the finitely generated free groups”, J. London Math. Soc., 29 (1954), 428–434 | DOI | MR | Zbl

[4] Eidelkind D. I., “O konechno porozhdennykh normalnykh delitelyakh polinilpotentnykh grupp”, Sib. matem. zhurn., 9:1 (1968), 236–239 | MR | Zbl

[5] Baumslag G., “Some theorems on the free groups of certain product varieties”, J. Combin. Theory, 2 (1967), 77–99 | DOI | MR | Zbl

[6] Ol'shanskii A. Yu., “Distortion functions for subgroups”, Geometric group theory down under (Canberra, Australia, 1996), Walter de Gruyter, 1999, 281–291 | MR

[7] Kholl F., “Nilpotentnye gruppy”, Matematika. Sb. per., 12:1 (1968), 3–36 | MR

[8] Neiman Kh., Mnogoobraziya grupp, Mir, M., 1969 | MR

[9] Shmelkin A. L., “Spleteniya i mnogoobraziya grupp”, Izv. AN SSSR. Ser. matem., 29:1 (1965), 149–170 | MR | Zbl

[10] Wolf J. A., “Growth of finitely generated solvable groups and curvature of Riemannian manifolds”, J. Differential Geom., 2 (1968), 421–446 | MR | Zbl

[11] Bass H., “The degree of polynomial growth of finitely generated nilpotent groups”, Proc. London Math. Soc. (3), 25 (1972), 603–614 | DOI | MR | Zbl

[12] Maltsev A. I., “O gruppakh konechnogo ranga”, Matem. sb., 22:2 (1948), 351–352 | MR | Zbl