Geodesic
A finitely generated complete group
Izvestiya. Mathematics, Tome 29 (1987) no. 2, pp. 233-277 
V. S. Guba. A finitely generated complete group. Izvestiya. Mathematics, Tome 29 (1987) no. 2, pp. 233-277. http://geodesic.mathdoc.fr/item/IM2_1987_29_2_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

An example of a $2$-generated complete group (different from the identity) with unique extraction of roots is constructed. An example is indicated, in passing, of a noncyclic $2$-generated group in which every element is conjugate to some power of a fixed element. It is proved that there are continuum many nonisomorphic such examples. The proof is based on a method developed in several papers of A. Yu. Ol'shanskii. Bibliography: 8 titles.

[1] Olshanskii A. Yu., “O teoreme Novikova–Adyana”, Matem. sb., 118:2 (1982), 203–235 | MR

[2] Olshanskii A. Yu., “Gruppy ogranichennogo perioda s podgruppami prostogo poryadka”, Algebra i logika, 21:5 (1982), 553–618 | MR

[3] Kourovskaya tetrad: Nereshennye voprosy teorii grupp, 8-e izdanie, Novosibirsk, 1982

[4] Adyan S. I., Problema Bernsaida i tozhdestva v gruppakh, Nauka, M., 1975 | MR | Zbl

[5] Lindon R., Shupp P., Kombinatornaya teoriya grupp, Nauka, M., 1980

[6] Olshanskii A. Yu., “O gruppakh s tsiklicheskimi podgruppami”, Dokl. AN Bolgarii, 32:9 (1979), 1165–1166 | MR

[7] Kargapolov M. I., Merzlyakov Yu. I., Osnovy teorii grupp, Nauka, M., 1982 | MR | Zbl

[8] Rips E., “Generalized small cancellation theory and applications. I. The word problem”, Israel J. Math., 41 (1982), 1–146 | DOI | MR | Zbl