A discreteness criterion for subgroups of PU(2,1) with screw parabolic elements

Abstract We establish a discreteness criterion for Γ ⊂ PU ( 2 , 1 ) with a screw parabolic element g , which can be viewed as a generalization of Shimizu's 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.