Talk:Torsion-free solvable hyperbolic groups (Ex)

Every hyperbolic group is virtually cyclic or contains the free group of rank two as a subgroup. Since the free group of rank two is not solvable and subgroups of solvable groups are solvable, a solvable hyperbolic group is virtually cyclic. The only torsion-free, virtually cyclic groups are the trivial group and the integers.