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Embedding free Burnside groups in finitely presented groups. (English) Zbl 1100.20031

The author constructs an embedding of a free Burnside group \(B(m,n)\) of odd exponent \(n>2^{48}\) into a finitely presented group with certain special properties.
One application of this embedding is a new simpler construction of finitely presented non-amenable groups without non-cyclic free subgroups. Earlier such groups [which solve Grigorchuk’s Problem 8.7 in The Kourovka Notebook (2006; Zbl 1084.20001)] were constructed by A. Yu. Ol’shanskii and M. V. Sapir [Publ. Math., Inst. Hautes Étud. Sci. 96, 43–169 (2002; Zbl 1050.20019)] using a different method. Another application is a construction of so-called weakly finitely presented groups of sufficiently large odd exponent \(n\) which are not locally finite.
Previously the author constructed weakly finitely presented periodic groups which are not locally finite [Contemp. Math. 296, 139–154 (2002; Zbl 1026.20024)], but those groups were of unbounded exponent. The proofs rely on the author’s technique of HNN-extensions for groups of large odd exponent [developed in Math. Appl., Dordr. 555, 39–53 (2003; Zbl 1042.20024)].

MSC:

20F50 Periodic groups; locally finite groups
20F05 Generators, relations, and presentations of groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F06 Cancellation theory of groups; application of van Kampen diagrams
20E07 Subgroup theorems; subgroup growth
20F38 Other groups related to topology or analysis
43A07 Means on groups, semigroups, etc.; amenable groups
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[2] Hausdorff F. (1914) Grundzüge der Mengenlehre, Leipzig · JFM 45.0123.01
[11] Kourovka Notebook: Unsolved problems in group theory, 8th edn, Novosibirsk, 1982
[13] Ol’shanskii, A. Yu.: Geometry of Defining Relations in Groups, Nauka, Moscow, 1989; English translation: Math. Appl. Soviet ser. 70, Kluwer Acad. Publ., Dordrecht, 1991
[16] Ivanov S.V.: On HNN-extensions in the class of groups of large odd exponent, In: Math. Appl. 555, Kluwer Acad. Publ., Dordrecht, 2003, pp. 39–54 · Zbl 1042.20024
[25] Adian S. I.: The Burnside Problem and Identities in Groups, Nauka, Moscow, 1975; English translation: Springer-Verlag, 1979
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