  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.
