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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     title = {On some algorithmic properties of hyperbolic groups},
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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/