Complex hyperbolic geometry of Chain links

Abstract We study the geometry of the complex hyperbolic triangle group normal upper Delta Subscript 4 comma normal infinity comma normal infinity semicolon normal infinity <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo>;</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msub> </mml:math> $\Delta_{4,\infty,\infty;\infty}$ acting on the complex hyperbolic plane bold upper H Subscript double struck upper C Superscript 2 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> $\mathbf{H}^2_{\mathbb C}$ and show that the 3-manifold at infinity of normal upper Delta Subscript 4 comma normal infinity comma normal infinity semicolon normal infinity <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo>;</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msub> </mml:math> $\Delta_{4,\infty,\infty;\infty}$ is the complement of the chain link 8 Subscript 1 Superscript 4 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msubsup> <mml:mn>8</mml:mn> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> </mml:math> $8^4_1$ in the 3-sphere. In particular, the four-cusped hyperbolic 3-manifold upper S cubed minus 8 Subscript 1 Superscript 4 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>−</mml:mo> <mml:msubsup> <mml:mn>8</mml:mn> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> </mml:math> $S^3-8^4_1$ admits a spherical CR-uniformisation. Since the ideal boundary of the Ford domain is an infinite-genus handlebody, we introduce a new technique to construct embedded topological disks that cap off the genus, ensuring their boundaries are compatible with the side-pairings. This surgical approach facilitates the determination of the fundamental group of the manifold at infinity and simplifies the computation compared to [ 14 ]. Based on this result and previous studies, we conjecture that the chain link upper C Subscript p <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> $C_p$ serves as an ancestor of the 3-manifold at infinity for the critical complex hyperbolic triangle group <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-typ