Work

Sobolev spaces and uniform boundary representations

Year: 2023

Type: article

Abstract: We prove uniform boundedness of certain boundary representations on appropriate fractional Sobolev spaces $W^{s,p}$ with $p>1$ for arbitrary Gromov hyperbolic groups. These are closed subspaces of $L^p$ and in particular Hilbert spaces in the case $p=2$. This construction allows us, for an appropriate choice of $p$, to approximate the trivial representation through uniformly bounded representations. This phenomenon does not have analogue in the setting of isometric representations whenever the hyperbolic group considered has the Property (T). The key is the introduction of a notion of metrically conformal operator on a metric space endowed with a conformal structure à la Mineyev and a metric analogue of the isomorphisms of Sobolev spaces induced by the Cayley transform.

Source: ePrints Soton (University of Southampton)

DOI: 10.48550/arxiv.2306.09999

Authors: Boucher, Kevin, Spakula, Jan

Institution: University of Southampton

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Engineering and Physical Sciences Research Council

Award: G4969991361