Involution and Commutator Length in PU(n,1)

RESUMO

We study decompositions of holomorphic isometries of complex hyperbolic space into holomorphic involutions. We prove that, for every n>=3, the involution length of PU(n,1) is 4. This improves the higher-dimensional upper bound 8 of Paupert--Will, as well as the projective upper bound 5 implied by Bünger's linear decomposition theorem, to the optimal value 4. Combined with the two-dimensional result of Paupert--Will, this shows that the involution length of PU(n,1) is 4 for every n>=2. The core of the proof is a three-involution decomposition theorem for square roots: every g in PU(n,1) has a square root of involution length at most 3, and this uniform bound is optimal. The key construction is carried out first on a two- or three-dimensional indefinite block and then completed in higher dimensions by a necessary and sufficient spectral pairing criterion on the positive-definite orthogonal complement. The four-factor lower bound is provided by complex reflections in a point. As a consequence, every element of PU(n,1) is a single commutator, and the two elements in the commutator representation can be simultaneously reversed by the same nontrivial holomorphic involution. This extends the two-dimensional single-commutator result of Paupert--Will to every n>=2 and thereby verifies Djoković's Conjecture A for the family PU(n,1), n>=2.