Geodesic
On some algorithmic properties of hyperbolic groups
Izvestiya. Mathematics , Tome 35 (1990) no. 1, pp. 145-163.

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For hyperbolic groups the author establishes the solvability of the algorithmic problems of extracting a root of an element, determining the order of an element, membership of a cyclic subgroup, and existence of a solution of an arbitrary quadratic equation. It is proved that every hyperbolic group has a finite presentation for which the word problem can be solved by Dehn's algorithm. The concept of a hyperbolic group was introduced by M. Gromov in a 1986 preprint. Bibliography: 8 titles. 
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I. G. Lysenok. On some algorithmic properties of hyperbolic groups. Izvestiya. Mathematics , Tome 35 (1990) no. 1, pp. 145-163. http://geodesic.mathdoc.fr/item/IM2_1990_35_1_a6/

[1] Gromov M., Hyperbolic groups, Preprint IHES, Paris, 1986

[2] Lysenok I. G., “O resheniyakh kvadratichnykh uravnenii v gruppakh s usloviem malogo sokrascheniya”, Matem. zametki, 43:5 (1988), 577–593 | MR

[3] Lindon R., Shupp P., Kombinatornaya teoriya grupp, Mir, M., 1980 | MR

[4] Lipschutz S., “An Extension of Greendlinger's results on the word problem”, Proc. Amer. Math. Soc., 15 (1964), 37–43 | DOI | MR | Zbl

[5] Lipschutz S., “On Greendlinger groups”, Comm. Pure and Appl. Math., 23:5 (1970), 743–747 | DOI | MR | Zbl

[6] Lipschutz S., “On the word problem and $T$-fourth groups”, Word problems, North-Holland, Amsterdam, 1973, 443–452 | MR

[7] Comerford L. P., “Quadratic equations over small concellation groups”, J. Algebra, 69:1 (1981), 175–185 | DOI | MR | Zbl

[8] Comerford L. P., Edmunds C. C, “Quadratic equations over free groups and free products”, J. Algebra, 68:2 (1981), 276–297 | DOI | MR | Zbl