V. M. Buchstaber, O. V. Karpov, and S. I. Tertychnyi initiated the study of the rotation number integer quantization effect for a class of dynamical systems on a torus that includes dynamical systems modeling the dynamics of the Josephson junction. Focusing on this effect, we initiate the study of a similar rotation number quantization effect for a class of groups acting on the circle, including Artin’s braid groups.