# A Discreteness Condition for Subgroups of $$\mathbf {PU}$$ PU (2,1)

A Discreteness Condition for Subgroups of $$\mathbf {PU}$$ PU (2,1) We establish a sufficient condition for a subgroup $$G=\langle A, B\rangle$$ G = ⟨ A , B ⟩ of $$\mathbf {PU}(2,1)$$ PU ( 2 , 1 ) to be discrete, where the pair of isometries (A, B) is $$\mathbb {C}$$ C -decomposable. This discreteness criterion is a complex hyperbolic version of Gilman’s result in Gilman (Contemp Math 211:261–267, 1997). Furthermore, we give an example to show that one can use our method to ascertain the discreteness of groups that do not pass the classical discreteness test on isometric spheres. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

# A Discreteness Condition for Subgroups of $$\mathbf {PU}$$ PU (2,1)

, Volume 19 (3) – May 20, 2019
21 pages

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Publisher
Springer Journals
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-019-00275-y
Publisher site
See Article on Publisher Site

### Abstract

We establish a sufficient condition for a subgroup $$G=\langle A, B\rangle$$ G = ⟨ A , B ⟩ of $$\mathbf {PU}(2,1)$$ PU ( 2 , 1 ) to be discrete, where the pair of isometries (A, B) is $$\mathbb {C}$$ C -decomposable. This discreteness criterion is a complex hyperbolic version of Gilman’s result in Gilman (Contemp Math 211:261–267, 1997). Furthermore, we give an example to show that one can use our method to ascertain the discreteness of groups that do not pass the classical discreteness test on isometric spheres.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: May 20, 2019

### References

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