Decomposition of Complex Hyperbolic Isometries by Involutions

Abstract

In a recent work, Basmajian and Maskit have investigated the problem of nding involution and commutator lengths of the isometry group of real space forms. In this thesis we aim to investigate the problem for isometry group of the complex hyperbolic space. A k-re ection of the n-dimensional complex hyperbolic space Hn C is an element in U(n; 1) with negative type eigenvalue , j j = 1, of multiplicity k+1 and positive type eigenvalue 1 of multiplicity n 􀀀 k. We prove that every element in SU(n) is a product of atmost ve involutions using which it can be shown that a holomorphic isometry of Hn C is a product of at most four involutions and a complex k-re ection, k 2. We also give a short proof of the well-known result that every holomorphic isometry of Hn C is a product of two anti-holomorphic involutions.