English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We investigate properties of some spherical fonctions defined on hyperbolicgroups using boundary representations on the Gromov boundary endowed with thePatterson-Sullivan measure class. We prove sharp decay estimates for sphericalfunctions as well as spectral inequalities associated with boundaryrepresentations. This point of view on the boundary allows us to view theso-called \emph{property RD} (also called \emph{Haagerup's inequality}) as aparticular case of a more general behavior of spherical functions on hyperbolicgroups. In particular, we give a constructive proof using a boundary unitaryrepresentation of a result due to de la Harpe and Jolissaint asserting thathyperbolic groups satisfy \emph{property RD}. Finally, we prove that the familyof boundary representations studied in this paper, which can be regarded as aone parameter deformation of the boundary unitary representation, are slowgrowth representations acting on a Hilbert space admitting a proper$1$-cocycle.

Spherical functions and rapid decay for hyperbolic groups