Discreteness Of Normalize Of Discrete Subgroups In PU(n, 1) And An Application

Let G be a nonelementary discrete subgroup of PU(n,1).In this paper,we present a dimension condition which can be used to check whether the normalizer N of G is discrete. We prove that N is discrete if and only if the dimension of complex hyperbolic space generated by L(G) is n.As a consequence of the result we prove that if G is non-elementary discrete subgroup of PU(n,1) and doesn't contain elliptic element,the volume of H_C~n/G is finite,then the isometry group of H_C~n/G is finite group.