HYPERBOLIC GROUPS AND THEIR QUOTIENTS OF BOUNDED EXPONENTS

In 1987, Gromov conjectured that for every non-elementary hyperbolic g roup G there is an n = n(G) such that the quotient group G/G(n) is inf inite. The article confirms this conjecture. In addition, a descriptio n of finite subgroups of G/G(n) is given, it is proven that the word a nd conjugacy problem are solvable in G/G(n) and that boolean AND(k = 1 )(infinity) G(k) = {1}. The proofs heavily depend upon prior authors' results on the Gromov conjecture for torsion free hyperbolic groups an d on the Burnside problem for periodic groups of even exponents.