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

TL;DR

This paper studies the geometry of the complex hyperbolic bidisk, focusing on isometries and Dirichlet domains, revealing that certain cyclic subgroup actions produce Dirichlet domains with exactly two sides.

Contribution

It characterizes the Dirichlet domains of cyclic loxodromic isometries in the complex hyperbolic bidisk, showing they have two sides, a novel geometric insight.

Findings

Dirichlet domains have two sides for cyclic loxodromic actions

Analysis of isometries in the complex hyperbolic bidisk

Contribution to understanding geometric structures in complex hyperbolic 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.