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Discreteness criteria for subgroups in complex hyperbolic space. (English) Zbl 1009.32016

The authors study discreteness criteria for subgroups of U(1,n;) in complex hyperbolic space H n .

The main result in this paper is as follows: A nonelementary subgroup G of U(1,n;) in complex hyperbolic space H n with Condition A is discrete if and only if every two generator subgroup is discrete.

Here we say that G satisfies Condition A provided that G has no sequence {g j } of distinct elements of finite order such that Card(fix(g j ))= in H n and g j I as j.

They also prove that if a nonelementary subgroup G of U(1,n;) contains a sequence {g j } of distinct elements with Card(fix(g j )H n ) and g j I as j, then G contains a non-discrete, nonelementary two generator subgroup.

32Q45Hyperbolic and Kobayashi hyperbolic manifolds
32M99Complex spaces with a group of automorphisms
22E99Lie groups