On some algorithmic properties of hyperbolic groups

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.
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