Abstract
This paper is a survey of the work of the authors [21], [2], [22], with a new application to Diophantine approximation in the Heisenberg group. The Heisenberg group, endowed with its Carnot-Caratheodory metric, can be seen as the space at infinity of the complex hyperbolic space (minus one point). The rational approximation on the Heisenberg group can be interpreted and developed using arithmetic subgroups of SU (n, 1). In the appendix, the case of hyperbolic surfaces is developed by Jouni Parkkonen and the second author.
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