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<p>In this work we find generators for the stabilizer subgroup of the point at infinity, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbf {q_\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">q</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>, of the Picard modular groups <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(U(3, 1; \mathcal {O}_d)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>;</mo> <msub> <mi mathvariant="script">O</mi> <mi>d</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(d = 2, 7, 11\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>11</mn> </mrow> </math></EquationSource> </InlineEquation>. In the case of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(d=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, the generating set consists of two Heisenberg rotations and three Heisenberg translations. In the cases of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(d=7\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(d=11\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>11</mn> </mrow> </math></EquationSource> </InlineEquation> the generating sets consist of two Heisenberg rotations and two Heisenberg translations. </p>

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Stabilizer Subgroup of the Point \(\mathbf {q_\infty }\) of Euclidean Picard Modular Groups in Three Complex Dimensions

  • 期刊论文

  • 发表日期：

    2025年6月9日

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  • Ehsan Rasoulian,
  • Majid Heydarpour

摘要

In this work we find generators for the stabilizer subgroup of the point at infinity, \(\mathbf {q_\infty }\) q ∞ , of the Picard modular groups \(U(3, 1; \mathcal {O}_d)\) U ( 3 , 1 ; O d ) where \(d = 2, 7, 11\) d = 2 , 7 , 11 . In the case of \(d=2\) d = 2 , the generating set consists of two Heisenberg rotations and three Heisenberg translations. In the cases of \(d=7\) d = 7 and \(d=11\) d = 11 the generating sets consist of two Heisenberg rotations and two Heisenberg translations.