In this paper we study discreteness of complex hyperbolic triangle groups of type <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_981_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="126" /> </InlineMediaObject> <EquationSource Format="TEX">$[m,m,0; n_1, n_2, 2]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>m</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mn>0</mn> <mo>;</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_981_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$n_1, n_2, 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> in complex geodesics with pairwise distances m, m, 0. For fixed m, the parameter space of such groups is of real dimension one. We determine the possible orders for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_981_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$n_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>n</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_981_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$n_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>n</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and also intervals in the parameter space that correspond to discrete and non-discrete triangle groups.

Complex hyperbolic triangle groups of type $[m,m,0;n_1,n_2,2]$

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2025年2月7日

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Sam Povall,
Anna Pratoussevitch

摘要

In this paper we study discreteness of complex hyperbolic triangle groups of type $[m,m,0; n_1, n_2, 2]$ $[m, m, 0; n_{1}, n_{2}, 2]$ , i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders $n_1, n_2, 2$ $n_{1}, n_{2}, 2$ in complex geodesics with pairwise distances m, m, 0. For fixed m, the parameter space of such groups is of real dimension one. We determine the possible orders for $n_1$ $n_{1}$ and $n_2$ $n_{2}$ and also intervals in the parameter space that correspond to discrete and non-discrete triangle groups.

Complex hyperbolic triangle groups of type \([m,m,0;n_1,n_2,2]\)

摘要