The mathematical problem of group synchronization deals with the question of how to estimate unknown set of group elements from a set of their mutual relations. This problem appears as an important step in solving many real world problem such as Structure from Motion (SfM) in vision or pose graph optimization (estimating positions and orientations of mobile robots) in robotics. In this talk, we present a novel solution for synchronization over the class of the non-compact Cartan motion groups, which includes the special important case of rigid motions. Our method is based upon the idea of group contraction, which is a compactification process origin in relativistic mechanics. We show the construction of such a solution for synchronization, analyze some of its theoretical aspects, and illustrate numerically its advantages compared to some of the current state-of-the-art synchronization methods on both synthetic and real data.