In this talk we show that if a generic periodic bar-joint framework (with a fixed lattice representation) is vertex-redundantly rigid, in the sense that the deletion of any vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. Our proof is similar to the one of Tanigawa for finite frameworks, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property in dimension greater than 2. We work around this issue by using slight modifications of recent results of Kaszanitzy, Schulze and Tanigawa (2021). Secondly, it is non-trivial to show that the rigidity of periodic frameworks with few vertex orbits implies their global rigidity. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion between two equivalent d-dimensional realisations of a single graph in (2d)-dimensional space to periodic frameworks.
As an application of our result, we give a combinatorial characterisation of generic globally rigid periodic body-bar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly, Jordan and Whiteley (2013) regarding the global rigidity of generic finite body-bar frameworks.

This is joint work with Csaba Kiraly and Viktoria Kaszanitzky.