Searcharxiv⌕ Search

arXiv subjects

Jean-Claude Picaud

Publications and source records attributed to Jean-Claude Picaud.

4 recordsLinked to original sources
arXiv2023

Riesz operators and some spherical representations for hyperbolic groups

We introduce the Riesz operator in the context of Gromov hyperbolic groups in order to investigate a one parameter family of non unitary boundary Hilbertian representations of hyperbolic groups. We prove asymptotic Schur's relations, the latter being the main result of this paper. Up to normalization, the Riesz operator plays the role in the context of hyperbolic groups of the Knapp-Stein intertwiner for complementary series for Lie groups. Assuming the positivity of the Riesz operator, we define an analogue of complementary series for hyperbolic groups and prove their irreducibility.

math.GR↗
arXiv2016

Ergodic geometry for non-elementary rank one manifolds

Let $X$ be a Hadamard manifold, and $Γ$ a non-elementary discrete group of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/Γ$ to the behavior of the Poincar{é} series of $Γ$. Precisely, the aim of this paper is to extend the so-called theorem of Hopf-Tsuji-Sullivan -- well-known for manifolds of pinched negative curvature -- to the framework of rank one orbifolds. Moreover, we derive some important properties for $Γ$-invariant conformal densities supported on the geometric limit set of $Γ$.

math.DG↗
arXiv2009

On the growth of nonuniform lattices in pinched negatively curved manifolds

We study the relation between the exponential growth rate of volume in a pinched negatively curved manifold and the critical exponent of its lattices. These objects have a long and interesting story and are closely related to the geometry and the dynamical properties of the geodesic flow of the manifold .

math.DS↗