Recent zbMATH articles in MSC 57S25https://www.zbmath.org/atom/cc/57S252021-04-16T16:22:00+00:00WerkzeugNotes on functions of hyperbolic type.https://www.zbmath.org/1456.430022021-04-16T16:22:00+00:00"Monod, Nicolas"https://www.zbmath.org/authors/?q=ai:monod.nicolasSummary: 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 \(\mathbf{SU}(1,n)\) and of its infinite-dimensional kin \(\text{Is}(\mathbf{H}_{\mathbf{C}}^{\infty})\). We further classify all the self-representations of \(\text{Is}(\mathbf{H}_{\mathbf{C}}^{\infty})\) that satisfy a compatibility condition for the subgroup \(\text{Is}(\mathbf{H}_{\mathbf{R}}^{\infty})\). 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.