Decomposition of complex hyperbolic isometries by involutions

A k-reflection of the n  -dimensional complex hyperbolic space HCn is an element in U(n,1)U(n,1) with negative type eigenvalue λ  , |λ|=1|λ|=1, of multiplicity k+1k+1 and positive type eigenvalue 1 of multiplicity n−kn−k. We prove that a holomorphic isometry of HCn is a product of at most four involutions and a complex k  -reflection, k≤2k≤2. Along the way, we prove that every element in SU(n)SU(n) is a product of four or five involutions according as n≢2mod4 or n≡2mod4. We also give a short proof of the well-known result that every holomorphic isometry of HCn is a product of two anti-holomorphic involutions.