On The Geometry Of A Subgroup Of The Eisenstein-picard Modular Group Of Index Six

Complex hyperbolic geometry is the study of the invariant geometry of a complex hy-perbolic space under the action of its isometric transformation group.In complex hyper-bolic geometry,the presentation of a discrete subgroup can be obtained by constructing the fundamental domain of this subgroup action on the complex hyperbolic space and then us-ing Poincarépolyhedron theorem.The Eisenstein-Picard modular group PU(2,1;Z[ω])is a very important discrete subgroup of PU(2,1),and particularly,it is an arithmetic lattice.Falbel and Parker studied the geometry of the Eisenstein-Picard modular group on H_C~2,gave a presentation of the group by constructing a fundamental domain for the action on H_C~2.This paper mainly studied the geometry of a subgroup H of the Eisenstein-Picard modu-lar group PU(2,1;Z[ω])of index six,where H contains a representation of the fundamental group of the complement of the figure eight knot into PU(2,1),and H is a normal sub-group of a index three subgroup of the Eisenstein-Picard modular group.According to the coset decomposition of H in Eisenstein-Picard modular group PU(2,1;Z[?]),we can give a fundamental domain of H acting on the complex hyperbolic plane.By applying Poincarépolyhedron theorem,we obtain a presentation of H.