The geometry of the Gauss–Picard modular group

Abstract

We give a construction of a fundamental domain for PU(2, 1, Z[i]), that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers Z[i]. We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.

New construction of fundamental domains for certain Mostow groups

“…As in the arguments in Section 7.8 of [Falbel et al 2011], we see that there are three regions where the first (respectively second and third) of these quantities as above is the smallest tessellate around the face 1 , S −1 I 1 and S −1 tessellate around the face Ᏺ 234 . The cycle transformation corresponding to the face Ᏺ 234 is (S −1 I 1 ) 3 and n = 6, m = 1.…”

“…As described in [Falbel et al 2011], we will discuss the triangular face with the vertices z 2 , z 3 , z 4 on the intersection Ꮾ c ∩ S −1 (Ꮾ c ) in terms of two slice s-parameters. We give the details for this face on Ꮾ c ∩ S −1 (Ꮾ c ) and the others follow similarly.…”

“…More recently, there has been a renewed interest in the construction of fundamental domains [Deraux et al 2005;Falbel and Parker 2006;Falbel et al 2011;Parker 2006;Zhao 2011]. In particular, Deraux, Falbel and Paupert gave a new construction of fundamental domains for some of the groups considered in [Mostow 1980].…”

Complex hyperbolic lattices

“…The ring of integers [26], [27] has studied these groups in great detail, using a combination of arithmetic methods and algebraic geometry. The geometry of the group SU(H; O 3 ) has been studied by Falbel and Parker [17] and the geometry of SU(H; O 1 ) has been studied by by Francsics and Lax [18] and Falbel, Francsics and Parker [16]. There is an obvious generalisation of Picard modular groups to higher complex dimensions.…”

Generators of the Eisenstein–picard Modular Group

“…The boundary of the complex hyperbolic space is defined to be ∂H 2 C = P(V 0 ). [3] The Eisenstein-Picard modular group 423…”