# Conjugation Orbits of Loxodromic Pairs in SU(n,1)

## TL;DR

This paper classifies the conjugation orbits of pairs of loxodromic elements in the complex hyperbolic isometry group SU(n,1), providing a detailed understanding of their geometric and algebraic structure.

## Contribution

It introduces a classification of conjugation orbits of pairs of loxodromic elements in SU(n,1), advancing the understanding of their geometric configurations.

## Key findings

- Complete classification of conjugation orbits of loxodromic pairs
- Identification of invariants characterizing the orbits
- Insights into the geometric structure of complex hyperbolic isometries

## Abstract

Let ${\bf H}_{\mathbb C}^n$ be the $n$-dimensional complex hyperbolic space and ${\rm SU}(n,1)$ be the (holomorphic) isometry group. An element $g$ in ${\rm SU}(n,1)$ is called loxodromic or hyperbolic if it has exactly two fixed points on the boundary $\partial {\bf H}_{\mathbb C}^n$. We classify ${\rm SU}(n,1)$ conjugation orbits of pairs of loxodromic elements in ${\rm SU}(n,1)$.