  Lattice translation symmetry gives rise to a large class of "weak"
topological insulators (TIs), characterized by translation-protected gapless
surface states and dislocation bound states. In this work we show that space
group symmetries lead to constraints on the weak topological indices that
define these phases. In particular we show that screw rotation symmetry
enforces the Hall conductivity along the screw axis to be quantized in
multiples of the screw rank, which generally applies to interacting systems. We
further show that certain 3D weak indices associated with quantum spin Hall
effects (class AII) are forbidden by the Bravais-lattice and by glide or
even-fold screw symmetries. These results put a strong constraints on
candidates of weak TIs in the experimental and numerical search for topological
materials, based on the crystal structure alone.
