Geodesic
On the Hilbert–Poincaré series of graded algebras associated with groups
Sbornik. Mathematics, Tome 66 (1990) no. 1, pp. 211-229 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A description is given of groups for which the coefficients of the Hilbert–Poincaré series of the associated algebra have power growth. Examples of finitely generated $p$-groups for which the growth of the coefficients of the Hilbert–Poincaré series is of order $e^{\sqrt n}$ are constructed. The results are applied to the theory of degrees of growth of groups. Bibliography: 23 titles. 
@article{SM_1990_66_1_a11,
     author = {R. I. Grigorchuk},
     title = {On the {Hilbert{\textendash}Poincar\'e} series of graded algebras associated with groups},
     journal = {Sbornik. Mathematics},
     pages = {211--229},
     year = {1990},
     volume = {66},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1990_66_1_a11/}
}
TY  - JOUR
AU  - R. I. Grigorchuk
TI  - On the Hilbert–Poincaré series of graded algebras associated with groups
JO  - Sbornik. Mathematics
PY  - 1990
SP  - 211
EP  - 229
VL  - 66
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1990_66_1_a11/
LA  - en
ID  - SM_1990_66_1_a11
ER  - 
%0 Journal Article
%A R. I. Grigorchuk
%T On the Hilbert–Poincaré series of graded algebras associated with groups
%J Sbornik. Mathematics
%D 1990
%P 211-229
%V 66
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1990_66_1_a11/
%G en
%F SM_1990_66_1_a11
R. I. Grigorchuk. On the Hilbert–Poincaré series of graded algebras associated with groups. Sbornik. Mathematics, Tome 66 (1990) no. 1, pp. 211-229. http://geodesic.mathdoc.fr/item/SM_1990_66_1_a11/

[1] Kostrikin A. I., Shanin Yu. I., Lineinaya algebra i geometriya, Izd-vo MGU, M., 1980

[2] Lasard M., “Group analytiques $p$-adiquus”, Publ. Math. IHES., 1965, no. 26, 389–603

[3] Babenko I. K, “Problemy rosta i ratsionalnosti v algebre i topologii”, UMN, 41:2 (1986), 95–142 | MR | Zbl

[4] Golod E. S, Shafarevich I. R., “O bashne polei klassov”, Izv. AN SSSR. Ser. matem., 28 (1964), 261–272 | MR | Zbl

[5] Kokh X., Teoriya Galua $p$-rasshirenii, Mir, M., 1973 | MR

[6] Golod E. S., “O nil-algebrakh i finitno-approksimiruemykh $p$-gruppakh”, Izv. AN SSSR. Ser. matem., 28 (1964), 273–276 | MR | Zbl

[7] Grigorchuk R. I., O filtratsii Tsassenkhauza v $p$-gruppakh, Tez. dokl. XIX Vsesoyuznoi algebr. konf., Izd-vo Lvovsk. un-ta, Lvov, 1987

[8] Passi F., Group rings and their augmentation ideals, Lect. Notes in Math., 715, 1979 | MR | Zbl

[9] Jennings S. A., “The structure of the group ring of $p$-group over a modular field”, Trans. Amer. Math. Soc., 50:1 (1941), 175–185 | DOI | MR | Zbl

[10] Gromov M., “Groups of polynomial growth and expanding maps.”, Publ. Math. IHES, 53 (1981), 53–73 | MR

[11] Grigorchuk R. I., “Stepeni rosta konechno-porozhdennykh grupp i teoriya invariantnykh srednikh”, Izv. AN SSSR. Ser. matem., 48 (1984), 939–985 | MR

[12] Grigorchuk R. I., “O stepenyakh rosta $p$-grupp i grupp bez krucheniya”, Matem. sb., 126(168) (1985), 194–214 | MR | Zbl

[13] Endryus G., Teoriya razbienii, Nauka, M., 1982 | MR

[14] Bereznyi A. E., “Diskretnye subeksponentsialnye gruppy”, Zap. nauchn. seminarov LOMI, 123 (1983), 155–166 | MR | Zbl

[15] Ramanujan S., Collected papers, Chelsea, N.-Y., 1962 | MR

[16] Tits J., “Free subgroups in linear groups”, J. Algebra, 20:2 (1972), 250–270 | DOI | MR | Zbl

[17] Cepp Zh.-P., Algebry Li i gruppy Li, Mir, M., 1969 | MR

[18] Milnor J., “Growth in finitely generated solvable groups”, J. Diff. Geometry, 2 (1968), 447–449 | MR | Zbl

[19] Wolf J., “Growth of finitely generated solvanle groups and curvature of Riemannien manifolds”, J. Diff. Geometry, 2 (1968), 421–446 | MR | Zbl

[20] Dries L., Wilkie A. J., “Gromov's theorem on groups of polynomial growth and elementary logic”, J. Algebra, 89:2 (1984), 349–374 | DOI | MR | Zbl

[21] Grigorchuk R. I., “K probleme Bernsaida o periodicheskikh gruppakh”, Funktsion. analiz i ego pril., 14:1, 53–54 | MR | Zbl

[22] Grigorchuk R. I., “O periodicheskikh gruppakh porozhdennykh konechnymi avtomatami”, Tez. dokl. XVIII Vsesoyuznoi algebr. konf., Izd-vo Shtiintsa, Kishinev, 1985, 138

[23] Gupta N., Sidki S., “On the Burnside problem for periodic groups”, Math. Z., 182 (1983.), 385–388 | DOI | MR | Zbl