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In structural rigidity, one studies frameworks of bars and joints inEuclidean space. Such a framework is an articulated structure consisting ofrigid bars, joined together at joints around which the bars may rotate. In thispaper, we will describe articulated motions of realisations of incidencegeometries that uses the terminology of graph of groups, and describe themotions of such a framework using group theory. Our approach allows to model avariety of situations, such as parallel redrawings, scenes, polytopes,realisations of graphs on surfaces, and even unique colourability of graphs.This approach allows a concise description of various dualities in rigiditytheory. We also provide a lower bound on the dimension of the infinitesimalmotions of such a framework in the special case when the underlying group is aLie group.

Structural rigidity and flexibility using graphs of groups