Abstract
The class of branch groups is defined (both in the abstract and in the profinite category). The relationship of this class with the class of extremal groups is established. Properties of the branch groups are investigated. Applications of the congruence property to the theory of profinite branch groups are indicated. The weak maximality of parabolic subgroups in branch groups is proved.
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References
R. I. Grigorchuk, “Just infinite branch groups,” in:New Horizons in Pro-p-Groups, Ed. M. du Sautoy, D. Segal, and A. Shalev, Birkhauser, Basel (2000).
R. I. Grigorchuk, “On extremal groups and branch groups,” in:Abstracts of Papers of the International Algebraic Conference in Memory of A. G. Kurosh [in Russian], Moscow (1998), pp. 163–165.
R. I. Grigorchuk, “Burnside’s problem on periodic groups,”Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.],14, No. 1, 53–54 (1980).
J. S. Wilson, “Groups with every proper quotient finite,”Proc. Camb. Phil. Soc.,69, 373–391 (1971).
J. S. Wilson, “On just infinite abstract and profinite groups,” in:New Horizons in Pro-p-Groups, Ed. M. du Sautoy, D. Segal, A. Shalev, Birkhauser, Basel (2000).
R. I. Grigorchuk, “Degrees of growth of finitely generated groups and the theory of invariant means,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],48, No. 5, 939–985 (1984).
R. I. Grigorchuk, “Degrees of growth ofp-groups and torsion-free groups,”Mat. Sb. [Math. USSR-Sb.],126, No. 2, 194–214 (1985).
N. Gupta and S. Sidki, “On the Burnside problem for periodic groups,”Math. Z.,182, 385–388 (1983).
C. V. Aleshin, “Finite automata and the Burnside problem for periodic groups,”Mat. Zametki [Math. Notes],11, No. 3, 319–328 (1972).
V. I. Sushchanskii, “Periodicp-groups of permutations and the unrestricted Burnside problem,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],247, No. 3, 557–561 (1979).
A. V. Rozhkov, “On the theory of groups of Aleshin type,”Mat. Zametki [Math. Notes],40, No. 5, 572–589 (1986).
A. V. Rozhkov, “Finiteness conditions in automorphism groups of trees,”Algebra i Logika [Algebra and Logic],37, No. 2, 568–605 (1998).
L. P. Bartholdi and R. I. Grigorchuk,On the Parabolic Subgroups and Hecke Algebras Related to Some Fractal Groups, Preprint of FIM, ETH-Zentrum, Zurich (1999).
R. I. Grigorchuk, W. N. Herfort, and P. A. Zaleskii, “The profinite completion of certain torsionp-groups,” in:Proc. of the International Algebraic Conference in Memory of A. G. Kurosh, Moscow (1998).
A. Shalev, “Subgroups structure, fractal dimension, and Kac-Moody algebras,” in:Groups and Geometries. Siena Conference, 1996, Ed. Lino di Martino, Birkhäuser (1998), pp. 163–176.
G. A. Margulis and G. A. Souifer, “Maximal subgroups of infinite index in finitely generated linear groups,”J. Algebra,69, No. 1, 1–23 (1981).
A. V. Rozhkov, “Centralizers of elements in a group of tree automorphisms,”Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.],57, No. 6, 82–105 (1993).
E. L. Pervova, “Dense subgroups of automorphism groups of trees” (to appear).
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Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 852–858, June, 2000.
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Grigorchuk, R.I. Branch groups. Math Notes 67, 718–723 (2000). https://doi.org/10.1007/BF02675625
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DOI: https://doi.org/10.1007/BF02675625