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Presentations for the Euclidean Picard modular groups

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Abstract

Mark and Paupert devised a general method for obtaining presentations for arithmetic non-cocompact lattices, \(\Gamma \), in isometry groups of negatively curved symmetric spaces. The method involves a classical theorem of Macbeath applied to a \(\Gamma \)-invariant covering by horoballs of the negatively curved symmetric space upon which \(\Gamma \) acts. In this paper, we will discuss the application of their method to the Picard modular groups, PU\((2,1;{\mathcal {O}}_{d})\), when \(d=2,11\), and obtain presentations for these groups, which completes the list of presentations for Picard modular groups whose entries lie in Euclidean domains, namely those with \(d=1,2,3,7,11\).

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References

  1. Chen, S., Greenberg, L.: Hyperbolic Spaces, in Contributions to Analysis, pp. 49–87. Academic Press, New York (1974)

    Book  Google Scholar 

  2. Falbel, E., Francsics, G., Parker, J.R.: The geometry of the Gauss–Picard modular group. Math. Ann. 349(2), 459–508 (2011)

    Article  MathSciNet  Google Scholar 

  3. Falbel, E., Parker, J.R.: The geometry of the Eisenstein–Picard modular group. Duke Math. J. 131(2), 249–289 (2006)

    Article  MathSciNet  Google Scholar 

  4. Goldman, W.M.: Complex Hyperbolic Geometry. Oxford University Press, Oxford Mathematical Monographs (1999)

  5. Kim, I., Parker, J.R.: Geometry of quaternionic hyperbolic manifolds. Math. Proc. Camb. Phil. Soc. 135, 291–320 (2003)

    Article  MathSciNet  Google Scholar 

  6. Macbeath, A.M.: Groups of homeomorphisms of a simply connected space. Ann. Math. 79(3), 473–488 (1964)

    Article  MathSciNet  Google Scholar 

  7. Magma Computational Algebra System. http://magma.maths.usyd.edu.au/magma/

  8. Mark, A., Paupert, J.: Presentations for cusped arithmetic hyperbolic lattices (2018). arXiv:1709.06691 (Preprint)

  9. Paupert, J., Will, P.: Real reflections, commutators and cross-ratios in complex hyperbolic space. Groups Geom. Dyn. 11, 311–352 (2017)

    Article  MathSciNet  Google Scholar 

  10. Polletta, D.: MATLAB code used to derive Euclidean Picard modular group presentations. https://github.com/DPolletta/Code-for-Euclidean-Picard-modular-group-derivations

  11. Swan, R.G.: Generators and relations for certain special linear groups. Adv. Math. 6, 1–77 (1971)

    Article  MathSciNet  Google Scholar 

  12. Zhao, T.: Generators for the Euclidean Picard modular groups. Trans. Am. Math. Soc. 364, 3241–3263 (2012)

    Article  MathSciNet  Google Scholar 

  13. Zink, T.: Über die Anzahl der Spitzen einiger arithmetischer Untergruppen unitärer Gruppen. Math. Nachr. 89, 315–320 (1979)

    Article  MathSciNet  Google Scholar 

Download references

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Correspondence to David Polletta.

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Author partially supported by National Science Foundation Grant DMS-1708463.

Appendices

Appendix A:\({\mathcal {O}}_2\)-rational point representatives

Depth

\({\mathcal {O}}_2\)-Rational points

1

\((0,0)\)

2

\((0,\sqrt{2})\)

3

\((\frac{2}{3}+\frac{i\sqrt{2}}{3},\frac{2}{3}\sqrt{2}),(\frac{2}{3}+\frac{2i\sqrt{2}}{3},\frac{2}{3}\sqrt{2}), (\frac{4}{3}+\frac{i\sqrt{2}}{3},0)\)

4

\((1,0),(1,\sqrt{2})\)

6

\((\frac{2}{3}+\frac{2i\sqrt{2}}{3},\frac{5}{3}\sqrt{2}),(\frac{4}{3}+\frac{i\sqrt{2}}{3},\sqrt{2}),(\frac{2}{3}+\frac{i\sqrt{2}}{3},\frac{5}{3}\sqrt{2})\)

8

\((0,\frac{\sqrt{2}}{2}),(0,\frac{3}{2}\sqrt{2}),(1,\frac{\sqrt{2}}{2}),(1,\frac{3}{2}\sqrt{2})\)

9

\((0,\frac{2}{3}\sqrt{2}), (0,\frac{4}{3}\sqrt{2}), (\frac{2}{3}+\frac{i\sqrt{2}}{3},0), (\frac{2}{3}+\frac{i\sqrt{2}}{3},\frac{4}{3}\sqrt{2}), (\frac{2}{3}+\frac{2i\sqrt{2}}{3},0),\)

 

\((\frac{2}{3}+\frac{2i\sqrt{2}}{3},\frac{4}{3}\sqrt{2}), (\frac{4}{3}+\frac{i\sqrt{2}}{3},\frac{2}{3}\sqrt{2}), (\frac{4}{3}+\frac{i\sqrt{2}}{3},\frac{4}{3}\sqrt{2}),\)

 

\((\frac{2}{9}+\frac{5i\sqrt{2}}{9},\frac{4}{3}\sqrt{2}), (\frac{4}{9}+\frac{i\sqrt{2}}{9},\frac{4}{9}\sqrt{2}), (\frac{8}{9}+\frac{2i\sqrt{2}}{9},\frac{16}{9}\sqrt{2}), (\frac{2}{9}+\frac{4i\sqrt{2}}{9},\frac{10}{9}\sqrt{2}), (\frac{10}{9}+\frac{2i\sqrt{2}}{9},\frac{4}{3}\sqrt{2}), (\frac{14}{9}+\frac{i\sqrt{2}}{9},\frac{10}{9}\sqrt{2})\)

11

\((\frac{2}{11}+\frac{3i\sqrt{2}}{11},\frac{8}{11}\sqrt{2}), (\frac{4}{11}+\frac{6i\sqrt{2}}{11},\frac{10}{11}\sqrt{2}), (\frac{8}{11}+\frac{i\sqrt{2}}{11},\frac{2}{11}\sqrt{2}), (\frac{10}{11}+\frac{4i\sqrt{2}}{11},\frac{4}{11}\sqrt{2}), (\frac{16}{11}+\frac{2i\sqrt{2}}{11},\frac{8}{11}\sqrt{2}),\)

 

\((\frac{2}{11}+\frac{8i\sqrt{2}}{11},\frac{18}{11}\sqrt{2}), (\frac{4}{11}+\frac{5i\sqrt{2}}{11},\frac{20}{11}\sqrt{2}), (\frac{6}{11}+\frac{2i\sqrt{2}}{11},\frac{6}{11}\sqrt{2}), (\frac{12}{11}+\frac{4i\sqrt{2}}{11},\frac{2}{11}\sqrt{2}), (\frac{14}{11}+\frac{i\sqrt{2}}{11},\frac{16}{11}\sqrt{2})\)

12

\((\frac{1}{3}+\frac{2i\sqrt{2}}{3},0), (\frac{1}{3}+\frac{2i\sqrt{2}}{3},\sqrt{2}), (\frac{1}{3}+\frac{2i\sqrt{2}}{3},\frac{16}{9}\sqrt{2}), (\frac{2}{9}+\frac{4i\sqrt{2}}{9},2\sqrt{2}), (\frac{1}{3}+\frac{i\sqrt{2}}{3},\frac{2}{3}\sqrt{2}), (\frac{1}{3}+\frac{i\sqrt{2}}{3},\frac{5}{3}\sqrt{2})\)

16

\((\frac{i\sqrt{2}}{2},0), (\frac{i\sqrt{2}}{2},\frac{\sqrt{2}}{2}), (\frac{i\sqrt{2}}{2},\sqrt{2}), (\frac{i\sqrt{2}}{2},\frac{3}{2}\sqrt{2}), (1+\frac{i\sqrt{2}}{2},0), (1+\frac{i\sqrt{2}}{2},\frac{\sqrt{2}}{2}), (1+\frac{i\sqrt{2}}{2},\sqrt{2}),\)

 

\((1+\frac{i\sqrt{2}}{2},\frac{3}{2}\sqrt{2})\)

Appendix B: Matrices sending \(\infty \) to \({\mathcal {O}}_{2}\)-rational points

$$\begin{aligned} I_0= & {} \begin{bmatrix} 0 &{} 0 &{} 1 \\ 0 &{} -1 &{} 0 \\ 1 &{} 0 &{} 0 \\ \end{bmatrix}, A_{2,1} =\begin{bmatrix} -1 &{} 0 &{} i\sqrt{2} \\ 0 &{} 1 &{} 0 \\ i\sqrt{2} &{} 0 &{} 1 \\ \end{bmatrix}, A_{3,1} = \begin{bmatrix} -1 &{} 0 &{} 0 \\ i\sqrt{2} &{} -1 &{} 0 \\ 1+i\sqrt{2} &{} i\sqrt{2} &{} -1 \\ \end{bmatrix},\\ A_{3,2}= & {} \begin{bmatrix} i\sqrt{2} &{} 0 &{} 1 \\ 2 &{} -1 &{} -i\sqrt{2} \\ 1-i\sqrt{2} &{} i\sqrt{2} &{} -1 \\ \end{bmatrix},\\ A_{3,3}= & {} \begin{bmatrix} -1+i\sqrt{2} &{} 2i\sqrt{2} &{} 2-i\sqrt{2} \\ 2-i\sqrt{2} &{} 1-2i\sqrt{2} &{} -2 \\ 1-i\sqrt{2} &{} -i\sqrt{2} &{} -1 \\ \end{bmatrix}, A_{4,1} = \begin{bmatrix} -1 &{} -2 &{} 2 \\ 2 &{} 3 &{} -2 \\ 2 &{} 2 &{} -1 \\ \end{bmatrix},\\ A_{4,2}= & {} \begin{bmatrix} -1+i\sqrt{2} &{} -2+i\sqrt{2} &{} 3 \\ 2 &{} 3 &{} -2-i\sqrt{2} \\ 2 &{} 2 &{} -1-i\sqrt{2} \\ \end{bmatrix},\\ A_{6,1}= & {} \begin{bmatrix} -3+i\sqrt{2} &{} 2-i\sqrt{2} &{} 3+3i\sqrt{2} \\ 2i\sqrt{2} &{} -1-2i\sqrt{2} &{} 4-i\sqrt{2} \\ 2+i\sqrt{2} &{} -2 &{} 1-2i\sqrt{2} \\ \end{bmatrix}, A_{6,2} =\begin{bmatrix} -3 &{} -2+2i\sqrt{2} &{} 2+2i\sqrt{2} \\ 2+2i\sqrt{2} &{} 3 &{} 2-2i\sqrt{2} \\ 2+i\sqrt{2} &{} 2 &{} 1-2i\sqrt{2} \\ \end{bmatrix},\\ A_{6,3}= & {} \begin{bmatrix} 1+2i\sqrt{2} &{} 2+2i\sqrt{2} &{} 2-2i\sqrt{2} \\ 2 &{} 3 &{} -2-2i\sqrt{2}\\ 2-i\sqrt{2} &{} 2-2i\sqrt{2} &{} -3 \\ \end{bmatrix}, A_{8,1} = \begin{bmatrix} -1 &{} 0 &{} i\sqrt{2} \\ 0 &{} 1 &{} 0 \\ 2i\sqrt{2} &{} 0 &{} 3 \\ \end{bmatrix}, A_{8,2} =\begin{bmatrix} -3 &{} 0 &{} i\sqrt{2} \\ 0 &{} 1 &{} 0 \\ 2i\sqrt{2} &{} 0 &{} 1 \\ \end{bmatrix},\\ A_{8,3}= & {} \begin{bmatrix} -1-i\sqrt{2} &{} -2-i\sqrt{2} &{} 1+i\sqrt{2} \\ 2i\sqrt{2} &{} 1+2i\sqrt{2} &{} -i\sqrt{2} \\ 2i\sqrt{2} &{} 2i\sqrt{2} &{} 1-2i\sqrt{2} \\ \end{bmatrix}, A_{8,4} =\begin{bmatrix} -3-i\sqrt{2} &{} -4-i\sqrt{2} &{} 3\\ 2i\sqrt{2} &{} 1+2i\sqrt{2} &{} -2-i\sqrt{2} \\ 2i\sqrt{2} &{} 2i\sqrt{2} &{} -1-i\sqrt{2} \\ \end{bmatrix},\\ A_{9,1}= & {} \begin{bmatrix} i\sqrt{2} &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 3 &{} 0 &{} -i\sqrt{2}\\ \end{bmatrix}, A_{9,2} = \begin{bmatrix} 2i\sqrt{2} &{} 0 &{} -1 \\ 0 &{} 1 &{} 0 \\ 3 &{} 0 &{} i\sqrt{2} \\ \end{bmatrix}, A_{9,3} =\begin{bmatrix} -1 &{} 0 &{} 0 \\ 2+i\sqrt{2} &{} 1 &{} 0 \\ 3 &{} 2-i\sqrt{2} &{} -1 \\ \end{bmatrix}, \\ A_{9,4}= & {} \begin{bmatrix} -1+2i\sqrt{2} &{} 2i\sqrt{2} &{} -i\sqrt{2} \\ 2+i\sqrt{2} &{} 3 &{} -2 \\ 3 &{} 2-i\sqrt{2} &{} -1+i\sqrt{2} \\ \end{bmatrix}, A_{9,5} =\begin{bmatrix} -2 &{} -2+2i\sqrt{2} &{} 3-2i\sqrt{2} \\ 2+2i\sqrt{2} &{} 5 &{} -6 \\ 3 &{} 2-2i\sqrt{2} &{} -2+3i\sqrt{2} \\ \end{bmatrix},\\ A_{9,6}= & {} \begin{bmatrix} -2+2i\sqrt{2} &{} -2-2i\sqrt{2} &{} -1-2i\sqrt{2} \\ 2+2i\sqrt{2} &{} -3 &{} -2 \\ 3 &{} -2+2i\sqrt{2} &{} -2+i\sqrt{2} \\ \end{bmatrix}, A_{9,7} = \begin{bmatrix} -3+i\sqrt{2} &{} 2-2i\sqrt{2} &{} 2 \\ 4+i\sqrt{2} &{} -5 &{} -2-2i\sqrt{2} \\ 3 &{} -4+i\sqrt{2} &{} -3-i\sqrt{2} \\ \end{bmatrix},\\ A_{9,8}= & {} \begin{bmatrix} -3+2i\sqrt{2} &{} -4+4i\sqrt{2} &{} 4-3i\sqrt{2} \\ 4+i\sqrt{2} &{} 7 &{} -6 \\ 3 &{} 4-i\sqrt{2} &{} -3+i\sqrt{2} \\ \end{bmatrix}, A_{9,9} = \begin{bmatrix} -3 &{} -2-i\sqrt{2} &{} 1+i\sqrt{2} \\ -2+i\sqrt{2} &{} -3 &{} 2 \\ 1+2i\sqrt{2} &{} 2i\sqrt{2} &{} -i\sqrt{2} \\ \end{bmatrix}, \end{aligned}$$
$$\begin{aligned} A_{9,10}= & {} \begin{bmatrix} -1 &{} 0 &{} i\sqrt{2} \\ i\sqrt{2} &{} -1 &{} 2 \\ 1+2i\sqrt{2} &{} i\sqrt{2} &{} 3-i\sqrt{2} \\ \end{bmatrix}, A_{9,11} = \begin{bmatrix} -4 &{} -2i\sqrt{2} &{} 1+2i\sqrt{2} \\ 2i\sqrt{2} &{} -1 &{} 2 \\ 1+2i\sqrt{2} &{} -2 &{} 2-i\sqrt{2} \\ \end{bmatrix},\\ A_{9,11}= & {} \begin{bmatrix} -4 &{} -2i\sqrt{2} &{} 1+2i\sqrt{2} \\ 2i\sqrt{2} &{} -1 &{} 2 \\ 1+2i\sqrt{2} &{} -2 &{} 2-i\sqrt{2} \\ \end{bmatrix}, A_{9,12} = \begin{bmatrix} 2+i\sqrt{2} &{} -2 &{} 1-2i\sqrt{2} \\ 2 &{} -1 &{} -2i\sqrt{2} \\ 1-2i\sqrt{2} &{} 2i\sqrt{2} &{} -4 \\ \end{bmatrix},\\ A_{9,13}= & {} \begin{bmatrix} 2+2i\sqrt{2} &{} -2+2i\sqrt{2} &{} -3 \\ 2-2i\sqrt{2} &{} 3 &{} 2+2i\sqrt{2} \\ 1-2i\sqrt{2} &{} 2 &{} 2+i\sqrt{2} \\ \end{bmatrix}, A_{9,14} = \begin{bmatrix} 1+3i\sqrt{2} &{} -2i\sqrt{2} &{} -4 \\ 2-3i\sqrt{2} &{} -1+2i\sqrt{2} &{} 4+2i\sqrt{2} \\ 1-2i\sqrt{2} &{} i\sqrt{2} &{} 3+i\sqrt{2} \\ \end{bmatrix},\\ A_{11,1}= & {} \begin{bmatrix} -1+i\sqrt{2} &{} -i\sqrt{2} &{} 1 \\ i\sqrt{2} &{} -1 &{} 0 \\ 3+i\sqrt{2} &{} -2 &{} -i\sqrt{2} \\ \end{bmatrix}, A_{11,2} = \begin{bmatrix} -2+i\sqrt{2} &{} 2 &{} 1 \\ 2i\sqrt{2} &{} 1-2i\sqrt{2} &{} 2-i\sqrt{2} \\ 3+i\sqrt{2} &{} -2-i\sqrt{2} &{} -1 \\ \end{bmatrix},\\ A_{11,3}= & {} \begin{bmatrix} -1 &{} 0 &{} 0 \\ 2+i\sqrt{2} &{} -1 &{} 0 \\ 3+i\sqrt{2} &{} -2+i\sqrt{2} &{} -1 \\ \end{bmatrix}, A_{11,4} =\begin{bmatrix} -2 &{} -2+2i\sqrt{2} &{} 3-i\sqrt{2} \\ 2+2i\sqrt{2} &{} 5 &{} -4-i\sqrt{2} \\ 3+i\sqrt{2} &{} 4-i\sqrt{2} &{} -3 \\ \end{bmatrix},\\ A_{11,5}= & {} \begin{bmatrix} -4 &{} -2i\sqrt{2} &{} 1-i\sqrt{2}\\ 4+2i\sqrt{2} &{} -1+2i\sqrt{2} &{} -2+i\sqrt{2} \\ 3+i\sqrt{2} &{} i\sqrt{2} &{} -1+i\sqrt{2} \\ \end{bmatrix}, A_{11,6} =\begin{bmatrix} 3i\sqrt{2} &{} 2-2i\sqrt{2} &{} -1-i\sqrt{2} \\ 2+2i\sqrt{2} &{} -1-2i\sqrt{2} &{} -2 \\ 3-i\sqrt{2} &{} -2 &{} i\sqrt{2} \\ \end{bmatrix},\\ A_{11,7}= & {} \begin{bmatrix} 1+3i\sqrt{2} &{} -4+3i\sqrt{2} &{} 1-3i\sqrt{2} \\ 2+i\sqrt{2} &{} -1+2i\sqrt{2} &{} -2i\sqrt{2} \\ 3-i\sqrt{2} &{} 4+2i\sqrt{2} &{} -4 \\ \end{bmatrix}, A_{11,8} = \begin{bmatrix} i\sqrt{2} &{} 0 &{} 1 \\ 2 &{} -1 &{} -i\sqrt{2} \\ 3-i\sqrt{2} &{} i\sqrt{2} &{} -1-i\sqrt{2} \\ \end{bmatrix},\\ A_{11,9}= & {} \begin{bmatrix} -2+i\sqrt{2} &{} -2i\sqrt{2} &{} 1-i\sqrt{2} \\ 4 &{} -3+2i\sqrt{2} &{} -2 \\ 3-i\sqrt{2} &{} -2+2i\sqrt{2} &{} -2 \\ \end{bmatrix}, A_{11,10} = \begin{bmatrix} -1+3i\sqrt{2} &{} -2-i\sqrt{2} &{} -3-i\sqrt{2} \\ 4-i\sqrt{2} &{} -1+2i\sqrt{2} &{} 2i\sqrt{2} \\ 3-i\sqrt{2} &{} 2i\sqrt{2} &{} 2i\sqrt{2} \\ \end{bmatrix},\\ A_{12,1}= & {} \begin{bmatrix} -1-i\sqrt{2} &{} i\sqrt{2} &{} 1 \\ -2+2i\sqrt{2} &{} 3 &{} -i\sqrt{2} \\ 2+2i\sqrt{2} &{} 2-2i\sqrt{2} &{} -1-i\sqrt{2} \\ \end{bmatrix}, A_{12,2} = \begin{bmatrix} -3 &{} 2+2i\sqrt{2} &{} 2+i\sqrt{2} \\ -2+2i\sqrt{2} &{} 3 &{} 2 \\ 2+2\sqrt{2} &{} 2-2i\sqrt{2} &{} 1-2i\sqrt{2} \\ \end{bmatrix},\\ A_{12,3}= & {} \begin{bmatrix} 1+i\sqrt{2} &{} -i\sqrt{2} &{} -1 \\ 2 &{} -1 &{} i\sqrt{2} \\ 2-2i\sqrt{2} &{} -2 &{} 1+i\sqrt{2} \\ \end{bmatrix}, A_{12,4} =\begin{bmatrix} 3+2i\sqrt{2} &{} -4+i\sqrt{2} &{} -3 \\ 2 &{} -1+2i\sqrt{2} &{} -2+i\sqrt{2} \\ 2-2i\sqrt{2} &{} 2+2i\sqrt{2} &{} 1+2i\sqrt{2} \\ \end{bmatrix},\\ A_{16,1}= & {} \begin{bmatrix} -1 &{} -i\sqrt{2} &{} 1 \\ 2i\sqrt{2} &{} -3 &{} -i\sqrt{2} \\ 4 &{} 2i\sqrt{2} &{} -1 \\ \end{bmatrix}, A_{16,2} =\begin{bmatrix} -1+i\sqrt{2} &{} 2-i\sqrt{2} &{} -1-2i\sqrt{2} \\ 2i\sqrt{2} &{} 1-2i\sqrt{2} &{} -4-i\sqrt{2} \\ 4 &{} -4-2i\sqrt{2} &{} -3+3i\sqrt{2} \\ \end{bmatrix}, \\ A_{16,3}= & {} \begin{bmatrix} -1+2i\sqrt{2} &{} 2 &{} -2-i\sqrt{2} \\ 2i\sqrt{2} &{} 1 &{} -2 \\ 4 &{} -2i\sqrt{2} &{} -1+2i\sqrt{2} \\ \end{bmatrix}, A_{16,4} = \begin{bmatrix} -1+3i\sqrt{2} &{} 4-3i\sqrt{2} &{} -1-3i\sqrt{2} \\ 2i\sqrt{2} &{} 1-2i\sqrt{2} &{} -2-i\sqrt{2} \\ 4 &{} -4-2i\sqrt{2} &{} -3+i\sqrt{2} \\ \end{bmatrix},\\ A_{16,5}= & {} \begin{bmatrix} -3 &{} 4+i\sqrt{2} &{} 3-i\sqrt{2} \\ 4+2i\sqrt{2} &{} -3-4i\sqrt{2} &{} -4-i\sqrt{2} \\ 4 &{} -4-2i\sqrt{2} &{} -3 \\ \end{bmatrix}, A_{16,6} = \begin{bmatrix} -3+i\sqrt{2} &{} 2+i\sqrt{2} &{} -1-3i\sqrt{2} \\ 4+2i\sqrt{2} &{} 1-2i\sqrt{2} &{} -4+3i\sqrt{2} \\ 4 &{} -2i\sqrt{2} &{} -1+3i\sqrt{2} \\ \end{bmatrix},\\ A_{16,7}= & {} \begin{bmatrix} -3+2i\sqrt{2} &{} 3i\sqrt{2} &{} -1-3i\sqrt{2} \\ 4+2i\sqrt{2} &{} 5 &{} -4+i\sqrt{2} \\ 4 &{} 4-2i\sqrt{2} &{} -3+2i\sqrt{2} \\ \end{bmatrix}, A_{16,8} = \begin{bmatrix} -3+3i\sqrt{2} &{} 4+i\sqrt{2} &{} -1-2i\sqrt{2} \\ 4+2i\sqrt{2} &{} 1-2i\sqrt{2} &{} -2+i\sqrt{2} \\ 4 &{} -2i\sqrt{2} &{} -1+i\sqrt{2} \\ \end{bmatrix} \end{aligned}$$

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Polletta, D. Presentations for the Euclidean Picard modular groups. Geom Dedicata 210, 1–26 (2021). https://doi.org/10.1007/s10711-020-00531-9

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