Geodesic
Certain balanced groups and 3-manifolds
Sbornik. Mathematics, Tome 188 (1997) no. 2, pp. 173-194 
Kim Ann Chi; A. I. Kostrikin. Certain balanced groups and 3-manifolds. Sbornik. Mathematics, Tome 188 (1997) no. 2, pp. 173-194. http://geodesic.mathdoc.fr/item/SM_1997_188_2_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Five series of groups with finite presentations are constructed. Their definition is based on the construction of some closed, compact, orientable 3-manifolds, so that these groups are balanced. The derived quotients of the groups are described. Almost all these groups are proved to be infinite; moreover, the linear groups $\operatorname {SL}(2,F)$ with $|F:{\mathbb Q}|\leqslant 6$ are involved in many of them. The relevant arguments are elementary, but the results obtained on balanced groups will be useful in further studies of 3-manifolds.

[1] Hempel J., $3$-manifolds, Ann. of Math. Stud., 86, Prinseton Univ. Press, Prineston, 1976 | MR | Zbl

[2] Holt D. F., Plesken W., “A cohomological criterion for a finitely presented group to be infinite”, J. London Math. Soc., 45:2 (1992), 469–480 | DOI | MR | Zbl

[3] Kim A. Ch., Kostrikin A. I., “Tri serii $3$-mnogoobrazii i ikh fundamentalnye gruppy”, Dokl. AN, 340:2 (1995), 158–160 | MR | Zbl

[4] Mennicke J., “Über Heegaarddiagramme von Geschlecht zwei mit endlicher Fundamentalgruppe”, Arch. Math. (Basel), 8 (1957), 192–198 | MR | Zbl

[5] Mennike I., Konechnye fundamentalnye gruppy trekhmernykh mnogoobrazii, Preprint, MIRAN, M., 1994, p. 1–15

[6] Milnor J., “Groups which act on $S^n$ without fixed points”, Amer. J. Math., 79 (1957), 623–630 | DOI | MR | Zbl

[7] Reidemeister K., “Kommutative Fundamentalgruppen”, Monatsh. Math. Phys., 43:1 (1935), 20–28 | DOI | MR

[8] Seifert H., Threlfall W., Lehrbuch der Topologie, Teubner, Leipzig, 1934 | Zbl

[9] Seifert H., Threlfall W., “Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bevegungsgruppen des dreidimensionalen sphärischen Raumes”, Math. Ann., 104 (1930–1931), 1–70 | MR | Zbl

[10] Wilson J. S., Zelmanov E. I., “Identities for Lie algebras of pro-$p$-groups”, J. Pure Appl. Algebra, 81 (1992), 103–109 | DOI | MR | Zbl