J Geom Anal (2012) 22:1–11 DOI 10.1007/s12220-010-9188-2 The Moduli Space of Points in the Boundary of Complex Hyperbolic Space Heleno Cunha · Nikolay Gusevskii Received: 25 April 2010 / Published online: 9 November 2010 © Mathematica Josephina, Inc. 2010 Abstract We consider the space M(n, m) of ordered m-tuples of distinct points in the boundary of complex hyperbolic n-space, H n C , up to its holomorphic isometry group PU(n, 1). An important problem in complex hyperbolic geometry is to con- struct and describe the moduli space for M(n, m). In particular, this is motivated by the study of the deformation space of complex hyperbolic groups generated by loxo- dromic elements. In the present paper, we give the complete solution to this problem. Keywords Complex hyperbolic space · Invariants · Gram matrix Mathematics Subject Classification (2000) 32H20 · 20H10 · 22E40 · 57S30 · 32G07 · 32C16 Introduction The study of the deformation space of groups in PU(n, 1) generated by loxodromic elements is one of the important problems in complex hyperbolic geometry. These groups, in some sense, are generic, and, moreover, if they are purely loxodromic then they are discrete; see Goldman [11]. Basic results related to the study of purely lox- odromic groups are contained in Parker–Platis [19]. There they used some invariants Communicated by Marco Abate. H. Cunha is supported by CNPq. N. Gusevskii is supported by CNPq and FAPEMIG. H. Cunha · N. Gusevskii ( ) Departamento de Matemática, Universidade Federal de Minas Gerais, 30123-970 Belo Horizonte, MG, Brazil e-mail: nikolay@mat.ufmg.br H. Cunha e-mail: cunha@mat.ufmg.br