We prove the group configuration theorem in simple theories, a very abstract result reconstructing a group (action) from a certain independence-theoretic configuration of points, and argue that such a result gives rise to 'geometric simplicity theory' (analogues of methods and results of geometric stability theory). The proof involves studying the behaviour of multivalued algebraic structures like polygroups and polyspaces, a development of the theory of independence for almost hyperimaginaries, and a sophisticated blowup procedure. Some of the corollaries of the group configuration theorem we obtain include finding the group associated to a polygroup in a simple theory, interpreting a vector space over a finite field inside a one-based w-categorical theory of SU-rank 1, and showing how pseudolinearity implies one-basedness under the assumption of w-categoricity.