Geodesic
Groups with Periodic Defining Relations
Matematičeskie zametki, Tome 83 (2008) no. 3, pp. 323-332 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the paper, the solvability of the word problem and the conjugacy problem is proved for a wide class of finitely presented groups defined by periodic defining relations of a sufficiently large odd degree. In the proof, we use a certain simplified version of the classification of periodic words and transformations of these words, which was presented in detail in the author's monograph devoted to the well-known Burnside problem. The result is completed with the proof of an interesting result of Sarkisyan on the existence of a group, given by defining relations of the form $E_i^2=1$, for which the word problem is unsolvable. This result was first published in abstracts of papers of the 13th All-Union Algebra Symposium in Gomel in 1975.
Keywords: finitely presented group, periodic defining relations, unsolvable conjugacy problem, unsolvable word problem, reduced word. 
@article{MZM_2008_83_3_a0,
     author = {S. I. Adian},
     title = {Groups with {Periodic} {Defining} {Relations}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {323--332},
     year = {2008},
     volume = {83},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2008_83_3_a0/}
}
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S. I. Adian. Groups with Periodic Defining Relations. Matematičeskie zametki, Tome 83 (2008) no. 3, pp. 323-332. http://geodesic.mathdoc.fr/item/MZM_2008_83_3_a0/

[1] S. I. Adian, The Burnside Problem and Identities in Groups, Ergeb. Math. Grenzgeb., 95, Springer-Verlag, Berlin–Heidelberg–New York, 1979 | MR | MR | Zbl | Zbl

[2] S. I. Adyan, V. G. Durnev, “Algoritmicheskie problemy dlya grupp i polugrupp”, UMN, 55:2 (2000), 3–94 | MR | Zbl

[3] O. A. Sarkisyan, Problema ravenstva slov dlya nekotorykh klassov grupp i polugrupp, Dis. ...kand. fiz.-matem. nauk, MGU, M., 1983