Notes on functions of hyperbolic type

TL;DR

This paper explores functions of hyperbolic type in complex hyperbolic spaces, establishing new deformation families of representations and classifying self-representations with specific compatibility conditions.

Contribution

It introduces the complex case for hyperbolic functions, constructs non-trivial deformation families of representations, and classifies self-representations satisfying certain subgroup compatibility.

Findings

Existence of non-trivial deformation families of representations of SU(1,n)

Classification of self-representations of Isom(H^ Infty_C) with subgroup compatibility

Translation lengths and Cartan arguments determine each other in these representations

Abstract

Functions of hyperbolic type encode representations on real or complex hyperbolic spaces, usually infinite-dimensional. These notes set up the complex case. As applications, we prove the existence of a non-trivial deformation family of representations of SU(1,n) and of its infinite-dimensional kin Isom(H^\infty_C). We further classify all the self-representations of Isom(H^\infty_C) that satisfy a compatibility condition for the subgroup Isom(H^\infty_R). It turns out in particular that translation lengths and Cartan arguments determine each other for these representations. In the real case, we revisit earlier results and propose some further constructions.