Uniqueness theorem for completely non-degenerate $B$-groups

We show that a completely non-degenerate $B$-group is uniquely determined by its factor: two such groups with conformally equivalent factors are Möbius conjugate. A similar property is inherent to the quasi-Fuchsian groups but not to degenerate $B$-groups. We also study the factor of a $B$-group as a triple: the main factor, the marked characteristic complex, and a homotopy class of maps of the first to the second one.