The geometry of the Eisenstein-Picard modular group. (English) Zbl 1109.22007
The paper studies the Eisenstein-Picard modular group \(\Gamma= \text{PU}(2;1;Z(\omega))\) which is by definition the subgroup of \(\text{PU}(2,1)\) with entries from \(Z(\omega)\), where \(\omega=\tfrac 12(-1+i\sqrt 3)\) is a cube root of 1. \(\Gamma\) acts isometrically and properly discontinuously on the complex hyperbolic space \(\mathbb CH^2\), which is realized here as the Siegel domain in the complex projective space \(\mathbb CP^2\). The authors construct a remarkably simple fundamental domain of \(\Gamma\), a 4-simplex with one ideal vertex. The construction is analogous to that for PSL\((2;\mathbb Z)\) acting on \(\mathbb CH^1\) where the well-known fundamental domain is a 2-simplex with one ideal vertex. The orbifold \(\mathbb CH^2/ \Gamma\) has volume \(\pi^2/27\) which is conjectured to be the smallest volume of a cusped complex hyperbolic orbifold.
Reviewer: Rainer Schimming (Greifswald)
MSC:
| 22E40 | Discrete subgroups of Lie groups |
| 11F60 | Hecke-Petersson operators, differential operators (several variables) |
| 11F55 | Other groups and their modular and automorphic forms (several variables) |
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