Applications conformes {\`a} grande {\'e}chelle

TL;DR

This paper introduces a large scale conformal map concept between metric spaces, showing it influences growth and boundary dimensions in groups, with implications for geometric group theory and metric invariants.

Contribution

It defines large scale conformal maps for metric spaces and demonstrates their impact on growth and boundary dimensions in groups, extending prior concepts.

Findings

Volume growth exponent increases under such maps in nilpotent groups.

Conformal dimension of hyperbolic group boundaries increases under these maps.

{ extl} p-cohomology is key to understanding these invariants.

Abstract

Roughly speaking, let us say that a map between metric spaces is large scale conformal if it maps packings by large balls to large quasi-balls with limited overlaps. This quasi-isometry invariant notion makes sense for finitely generated groups. Inspired by work by Benjamini and Schramm, we show that under such maps, some kind of dimension increases: exponent of volume growth for nilpotent groups, conformal dimension of the ideal boundary for hyperbolic groups. A purely metric space notion of {\ell} p-cohomology plays a key role.