# Hyperbolic spaces, principal series and ${\rm O}(2,\infty)$

## TL;DR

The paper proves the non-existence of certain irreducible representations of hyperbolic space isometry groups into ${
m O}(2,
finite)$ for dimensions three and higher, contrasting with known representations for other parameters.

## Contribution

It establishes a new non-existence result for irreducible representations of ${
m PO}(1,n)^{ ext{circ}}$ into ${
m O}(2,inite)$, expanding understanding of representation theory of hyperbolic groups.

## Key findings

- No irreducible representation of ${
m PO}(1,n)^{ ext{circ}}$ into ${
m O}(2,inite)$ for $n  3.
- Existence of irreducible representations into ${
m O}(p,inite)$ for other values of $p$.
- Results influence the classification of group representations related to hyperbolic geometry.

## Abstract

We prove that there exists no irreducible representation of the identity component of the isometry group ${\rm PO}(1,n)$ of the real hyperbolic space of dimension $n$ into the group ${\rm O}(2,\infty)$, if $n\geq 3$. This is motivated by the existence of irreducible representations (arising from the spherical principal series) of ${\rm PO}(1,n)^{\circ}$ into the groups ${\rm O}(p,\infty)$ for other values of $p$.