On a criterion for the topological conjugacy of a quasisymmetric group to a group of affine transformations of $\mathbb R$

A new criterion for the quasisymmetric conjugacy of an arbitrary group of orientation-preserving quasisymmetric homeomorphisms of the real line to some group of affine transformations is put forward.
In the criterion proposed by Hinkkanen one requires the uniform boundedness of constants involved in the definition of a quasisymmetric transformation over all elements of the group. In the new criterion only the uniform boundedness of constants for each cyclic subgroup is required.