Decomposition of complex hyperbolic isometries by involutions

TL;DR

This paper demonstrates that complex hyperbolic space isometries can be decomposed into a small number of involutions and reflections, providing new insights into their algebraic structure.

Contribution

It establishes that holomorphic isometries of complex hyperbolic space can be expressed as products of at most four involutions and a complex reflection, and also characterizes involution decompositions in SU(n).

Findings

Any holomorphic isometry is a product of at most four involutions and a complex $k$-reflection.

Every element in ${ m SU}(n)$ is a product of four or five involutions depending on $n$.

Every holomorphic isometry can be written as a product of two anti-holomorphic involutions.

Abstract

A $k$-reflection of the $n$-dimensional complex hyperbolic space ${\rm H}_{\C}^n$ is an element in ${\rm U}(n,1)$ with negative type eigenvalue $\lambda$, $|\lambda|=1$, of multiplicity $k+1$ and positive type eigenvalue $1$ of multiplicity $n-k$. We prove that a holomorphic isometry of ${\rm H}_{\C}^n$ is a product of at most four involutions and a complex $k$-reflection, $k \leq 2$. Along the way, we prove that every element in ${\rm SU}(n)$ is a product of four or five involutions according as $n \neq 2 \mod 4$ or $n = 2 \mod 4$. We also give an easy proof of the result that every holomorphic isometry of ${\rm H}_{\C}^n$ is a product of two anti-holomorphic involutions.