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

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.