  This paper describes some generalizations of the results presented in the
book "Geometry of defining Relations in Groups" , of A.Yu.Ol'shanskii to the
case of non-cyclic torsion-free hyperbolic groups. In particular, it is proved
that for every non-cyclic torsion-free hyperbolic group, there exists a
non-Abelian torsion-free quotient group in which all proper subgroups are
cyclic, and the intersection of any two of them is not trivial.
