Real hyperbolic representations of PU(1,n)

TL;DR

The paper proves that for dimensions greater than one, the holomorphic isometry group of complex hyperbolic space cannot be non-elementarily represented in the isometry group of infinite-dimensional real hyperbolic space.

Contribution

It establishes a non-existence result for non-elementary representations of complex hyperbolic isometry groups into infinite-dimensional real hyperbolic isometry groups.

Findings

No non-elementary representations exist for n > 1

Results apply to holomorphic isometries of complex hyperbolic space

Implications for the structure of hyperbolic isometry groups

Abstract

It is shown that for n bigger than 1, the group of holomorphic isometries of the n dimensional complex hyperbolic space does not admit non-elementary representations into the group of isometries of the infinite dimensional real hyperbolic space.