  In structural rigidity, one studies frameworks of bars and joints in
Euclidean space. Such a framework is an articulated structure consisting of
rigid bars, joined together at joints around which the bars may rotate. In this
paper, we will describe articulated motions of realisations of incidence
geometries that uses the terminology of graph of groups, and describe the
motions of such a framework using group theory. Our approach allows to model a
variety 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 rigidity
theory. We also provide a lower bound on the dimension of the infinitesimal
motions of such a framework in the special case when the underlying group is a
Lie group.
