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Equidistant Hypersurfaces Of The Complex Bidisk $\mathbb{H}^2_{\mathbb{C}}\times \mathbb{H}^2_{\mathbb{C}}$

Krishnendu Gongopadhyay, Lokenath Kundu, Aditya Tiwari

TL;DR

The paper analyzes Dirichlet domains for cyclic subgroups of the complex hyperbolic bidisk $\mathbb{H}^2_{\mathbb{C}}\times\mathbb{H}^2_{\mathbb{C}}$ endowed with the product metric. It shows that when the generator $\\\gamma=(g_1,g_2)$ has loxodromic components and the base point lies on the invariant axes, the two bisectors $E(z,\\gamma(z))$ and $E(z,\\gamma^{-1}(z))$ bound a Dirichlet domain with exactly two faces, leveraging the Hadamard geometry and a notion of invisibility. The work combines an explicit description of the isometry group, the structure of equidistant hypersurfaces, and asymptotic boundary analysis via Busemann functions to obtain a sharp two-face result, extending analogous real-bidisk phenomena to the complex setting. The results illuminate tiling properties and group actions in complex hyperbolic products and highlight how product structure constrains Dirichlet domains in higher-rank symmetric spaces.

Abstract

We consider the isometries of the complex hyperbolic bidisk, that is, the product space $\mathbb{H}^2_{\mathbb{C}} \times \mathbb{H}^2_{\mathbb{C}} $, where each factor $ \mathbb{H}^2_{\mathbb{C}} $ denotes the complex hyperbolic plane. We investigate the Dirichlet domain formed by the action of a cyclic subgroup $(g_1, g_2)$, where each $g_i$ is loxodromic. We prove that such a Dirichlet domain has two sides.

Equidistant Hypersurfaces Of The Complex Bidisk $\mathbb{H}^2_{\mathbb{C}}\times \mathbb{H}^2_{\mathbb{C}}$

TL;DR

The paper analyzes Dirichlet domains for cyclic subgroups of the complex hyperbolic bidisk endowed with the product metric. It shows that when the generator has loxodromic components and the base point lies on the invariant axes, the two bisectors and bound a Dirichlet domain with exactly two faces, leveraging the Hadamard geometry and a notion of invisibility. The work combines an explicit description of the isometry group, the structure of equidistant hypersurfaces, and asymptotic boundary analysis via Busemann functions to obtain a sharp two-face result, extending analogous real-bidisk phenomena to the complex setting. The results illuminate tiling properties and group actions in complex hyperbolic products and highlight how product structure constrains Dirichlet domains in higher-rank symmetric spaces.

Abstract

We consider the isometries of the complex hyperbolic bidisk, that is, the product space , where each factor denotes the complex hyperbolic plane. We investigate the Dirichlet domain formed by the action of a cyclic subgroup , where each is loxodromic. We prove that such a Dirichlet domain has two sides.

Paper Structure

This paper contains 12 sections, 8 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Complex Hyperbolic Plane H2C and complex bidisc
  3. Hermitian Form and the Ball Model
  4. Metric on the Complex Bidisc
  5. Busemann Functions in Complex Hyperbolic Space
  6. Subspace with holomorphic curvature $-1$
  7. Isometry Group of complex bidisc
  8. Isometry group of the complex bidisk
  9. Equidistant hypersurface
  10. Accumulation Points of Level Sets in Complex Hyperbolic Space
  11. Proof of Theorem \ref{['mainth']}
  12. Proof of Theorem \ref{['mainth']}

Key Result

Theorem 1.1

Let $\gamma = (g_1, g_2) \in \mathrm{Isom}(\mathbb{H}^2_{\mathbb{C}} \times \mathbb{H}^2_{\mathbb{C}})$, where $g_1$ and $g_2$ are loxodromic isometries of $\mathbb{H}^2_{\mathbb{C}}$. Let $z_1, z_2 \in \mathbb{H}^2_{\mathbb{C}}$ be points lying on the invariant axes of $g_1$ and $g_2$, respectively

Theorems & Definitions (19)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Theorem 3.2
  • proof
  • ...and 9 more