A discreteness criterion for subgroups of with screw parabolic elements

We establish a discreteness criterion for Γ ⊂ PU ( 2 , 1 ) with a screw parabolic element g, which can be viewed as a generalization of Shimizuʹs lemma. Our method relies upon the geometry of Margulis region. It allows us to get rid of the restriction on rotation angle 2 π α of g in [4,7]. Besides, an explicit region precisely invariant under the stabilizer of the fixed point of g in Γ is given.