A minimal volume arithmetic cusped complex hyperbolic orbifold

Abstract

The sister of Eisenstein–Picard modular group is described explicitly in [10], whose quotient is a noncompact arithmetic complex hyperbolic 2-orbifold of minimal volume (see [16]). We give a construction of a fundamental domain for this group. A presentation of that lattice can be obtained from that construction, which relates to one of Mostow's lattices.

Mostow’s lattices and cone metrics on the sphere

“…Once again, when v ±2 and v ±3 coalesce we obtain z 2 = 0 and when v 0 , v 3 coalesce we have equation (14). Putting this together gives…”

“…Also, the group corresponding to (θ, φ) = (2π/3, π/6) is the so called "sister" of the Eisenstein-Picard modular group considered by Zhao [14]. The fundamental domains we construct are different from the ones constructed in [5] and [14].…”

New construction of fundamental domains for certain Mostow groups

“…We now give the generators as R = (JR 1 J . Recall that the generators R, S, T, I 1 arise from the side-pairing maps of a fundamental domain constructed in [Zhao 2011], we call them geometrical generators. So the group k may be rewritten as R, S, T, I 1 .…”

“…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].…”

“…When p = 3 the values of k that lead to a lattice are exactly those for which there is an integer l so that 1/k + 1/l = 1 6 (see also the table in [Parker 2009, page 27]). In [Zhao 2011] we constructed a fundamental domain for the case k = 6. In this paper, we consider the case k ≥ 7 and construct a fundamental domain whose shape is based on that of the domain for k = 6.…”