Global Rigidity of Periodic Graphs under Fixed-lattice Representations

In 1992, Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in ${\mathbb{R}}^d$. Jackson and Jordan confirmed in 2005 that these conditions are also sufficient in ${\mathbb{R}}^2$, giving a combinatorial characterization of graphs whose generic realizations in ${\mathbb{R}}^2$ are globally rigid. In this paper, we establish analogues of these results for infinite periodic frameworks under fixed lattice representations. Our combinatorial characterization of globally rigid generic periodic frameworks in ${\mathbb{R}}^2$ in particular implies toroidal and cylindrical counterparts of the theorem by Jackson and Jordan.