Commuting Isometries of the Complex Hyperbolic Space

TL;DR

This paper investigates the conditions under which two isometries of complex hyperbolic space commute and characterizes their centralizers, enhancing understanding of the symmetry structure of complex hyperbolic spaces.

Contribution

It provides a detailed analysis of commuting isometries in complex hyperbolic space and determines their centralizers, a novel contribution to geometric group theory.

Findings

Characterization of commuting isometries in complex hyperbolic space

Explicit description of centralizers of isometries

Insights into the symmetry group structure of $H^n$

Abstract

Let $H^n$ denote the complex hyperbolic space of dimension $n$. The group $U(n,1)$ acts as the group of isometries of $H^n$. In this paper we investigate when two isometries of the complex hyperbolic space commute. Along the way we determine the centralizers.