Periodic nonlocally finite (Burnside) groups of infinite period are studied. The first explicit example of such groups was proposed by S.V. Aleshin in 1972. His construction was generalized to AT-groups, i.e., tree automorphism groups. A number of known problems have been solved with the help of AT-groups. Up to now, in reality, only the class of $AT_{\omega}$-groups, i.e., the class of AT-groups over a sequence of cyclic groups of prime order, has been studied. In this paper, the class of $AT_{\Omega}$-groups, i.e., of AT-groups over a sequence of cyclic groups of arbitrary finite order, is studied. The difference between $AT_{\omega}$-groups and true $AT_{\Omega}$-groups was revealed by the solution of the Kourovka Problem 16.79. The study of the class of $AT_{\Omega}$-groups has allowed us to introduce a number of new notions. Now the $AT_{\omega}$-groups can be considered as elementary AT-groups by which the AT-groups over a sequence of periodic groups are saturated. We propose a strategy for studying such AT-groups and give promising directions of this kind of research.
Keywords: Burnside groups, residually finite group, finiteness conditions, AT-groups, trees, wreath product
Received November 11, 2021
Revised January 18, 2021
Accepted January 24, 2021
Funding Agency: This work was supported by the Scholarship Program of the Vladimir Potanin Foundation.
Alexander Vicktorovich Rozhkov, Dr. Phys.-Math. Sci., Prof., Kuban State University, Krasnodar, 350040 Russia, e-mail: ros@math.kubsu.ru
REFERENCES
1. Golod E.S. On nil-algebras and finitely approximable p-groups. In: Fourteen papers on logic, algebra, complex variables and topology, AMS Translations: Ser. 2, 1965, vol. 48, pp. 103–106. doi: 10.1090/trans2/048/06
2. Aleshin S.V. Finite automata and Burnside’s problem for periodic groups. Math. Notes, 1972, vol. 11, no. 3, pp. 199–203. doi: 10.1007/BF01098526
3. Sushchansky V.I. Periodic p-groups of permutations and the unrestricted Burnside problem. Sov. Math., Dokl., 1979, vol. 20, pp. 766–770.
4. Grigorchuk R.I. Burnside’s problem on periodic groups. Funct. Anal. Appl., 1980, vol. 14, no. 1, pp. 41–43. doi: 10.1007/BF01078416
5. Gupta N., Sidki S. Some infinite p-groups. Algebra Logika, 1983, vol. 22, no. 5, pp. 584–589.
6. Merzlyakov Yu.I. On infinite finitely generated periodic groups. Sov. Math., Dokl., 1983, vol. 27, pp. 169–172.
7. Rozhkov A.V. Theory of Aleshin type groups. Math. Notes, 1986, vol. 40, no. 5, pp. 827–836. doi: 10.1007/BF01159699
8. Grigorchuk R.I. Degrees of growth of finitely generated groups, and the theory of invariant means. Math. USSR-Izv., 1985, vol. 25, no. 2, pp. 259–300. doi: 10.1070/IM1985v025n02ABEH001281
9. Rozhkov A.V. Usloviya konechnosti v gruppakh avtomorfizmov derev’ev [Finiteness conditions in groups of tree automorphisms]. Doctor Sci. (Phys.–Math.) Dissertation, Krasnoyarsk: Kranoyarsk. Gos. Univ., 1997, 230 p.
10. Unsolved problems in group theory. The Kourovka notebook. No. 18, eds. V. D. Mazurov and E. I. Khukhro, Novosibirsk: Inst. Math. SO RAN Publ., 2014, 228 p. Available on: https://arxiv.org/pdf/1401.0300v10.pdf
11. Grigorchuk R.I. Branch groups. Math. Notes, 2000, vol. 67, no. 6, pp. 718–723. doi: 10.1007/BF02675625
Cite this article as: A.V. Rozhkov. AT-groups that are not AT-subgroups: Transition from $AT_{\omega}$-groups to $AT_{\Omega}$-groups, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 1, pp. 218–231.
