Generalized isometric spheres and fundamental domains for discrete subgroups of $PU(1,n; \mathbf {C})$

Abstract

Let $G$ be a discrete subgroup of $PU(1,n; \mathbf{C})$. For a boundary point $y$ of the Siegel domain, we define the generalized isometric sphere $I_y(f)$ of an element $f$ of $PU(1,n; \mathbf{C})$. By using the generalized isometric spheres of elements of $G$, we construct a fundamental domain $P_y(G)$ for $G$, which is regarded as a generalization of the Ford domain. And we show that the Dirichlet polyhedron $D(w)$ for $G$ with center $w$ convereges to $P_y(G)$ as $w \rightarrow y$. Some results are also found in [5], but our method is elementary.