Decomposition of complex hyperbolic isometries by 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.