On some factor groups of hyperbolic groups (Q1898565)

Consider a group \(\Gamma\) generated by a finite set \(\mathcal A\) and a left- invariant metric on \(\Gamma\) for which \(|g - h|= |g^{-1} h|\), where by \(|g|\) is meant the length of the shortest word in the alphabet \({\mathcal A}^{\pm 1}\), representing \(g\) belonging to \(\Gamma\). The Gromov product of two elements \(g, h \in \Gamma\) is the expression \((g \cdot h) = 1/2(|g|+ |h|- |g^{- 1} h|)\), and, by definition, the group \(\Gamma\) is \(\delta\)- hyperbolic if for some \(\delta \geq 0\) and \(\forall x, y, z \in \Gamma\) the inequality \((x \cdot y) \geq \min((x \cdot z)\), \((y \cdot z)) - \delta\) holds. The present paper is devoted to the following assertions. Theorem 1. For an arbitrary nonelementary hyperbolic group without torsion \(\Gamma\) and any sufficiently large odd number \(n \geq n_0 (\Gamma)\) there exists an infinite factor group \(\Gamma\) whose proper subgroups are cyclic groups of order that divides \(n\). Theorem 2. For an arbitrary nonelementary hyperbolic group without torsion \(\Gamma\) there exists an infinite factor group without torsion, all of whose proper subgroups are cyclic.