Modul: MAT772 Geometrie-Seminar

## Spherical functions on hyperbolic groups and property RD (Rapid Decay)

Dr. Adrien Boyer talk

**Date:** 27.03.19 **Time:** 15.45 - 18.00 **Room:** ETH HG G 43

We investigate properties of some spherical fonctions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson-Sullivan measure class. We prove sharp decay estimates for spherical functions as well as spectral inequalities associated with boundary representations. This point of view on the boundary allows us to view the so-called property RD (Rapid Decay, also called Haagerup's inequality) as a particular case of a more general behavior of spherical functions on hyperbolic groups. If time permits, I will try to explain the idea of a constructive proof using a boundary unitary representation of a result due to de la Harpe and Jolissaint asserting that hyperbolic groups satisfy property RD.