Boundary representations of hyperbolic groups: the log-Sobolev case

TL;DR

This paper investigates boundary representations of hyperbolic groups on a specific function space related to the logarithmic Laplacian, revealing their growth behavior and establishing related inequalities.

Contribution

It introduces the analysis of boundary representations on $W^{ ext{log},2}(oundary ext{Gamma})$, showing their non-uniform boundedness and precise growth rate, along with $L^p$ analogues.

Findings

Boundary representations grow with the square root of group element length.

They are not uniformly bounded on the function space.

Logarithmic Sobolev inequality is established for Ahlfors--David regular spaces.

Abstract

We study boundary representations of hyperbolic groups $\Gamma$ on the (compactly embedded) function space $W^{\log,2}(\partial\Gamma)\subset L^2(\partial\Gamma)$, the domain of the logarithmic Laplacian on $\partial\Gamma$. We show that they are not uniformly bounded, and establish their exact growth (up a multiplicative constant): they grow with the square root of the length of $g\in\Gamma$. We also obtain $L^p$--analogue of this result. Our main tool is a logarithmic Sobolev inequality on bounded Ahlfors--David regular metric measure spaces.