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Quasi-conformal VS quasi-isometric equivalence in spaces with controlled growth

Katrin FÄssler, Enrico Le Donne, Sebastiano Nicolussi Golo, Alessandro Ottazzi, Pierre Pansu

TL;DR

This work establishes a comprehensive QC→QI principle for spaces with controlled growth, centering on geodesic Lie groups equipped with left-invariant geodesic distances and Haar measure. It develops a capacity–based framework using Ferrand distances in hyperbolic and parabolic conformal types to bridge infinitesimal quasiconformality with large-scale geometry, enabling bi-bornologous behavior and quasi-isometries under broad hypotheses. In particular, it proves that any quasi-conformal map between simply connected nilpotent geodesic Lie groups is a quasi-isometry, and more generally that QC maps between geodesic Lie groups are quasi-isometries when the conformal type is hyperbolic or strictly parabolic; the liminal parabolic case presents counterexamples. The methodology integrates capacity estimates, isoperimetric inequalities at large scale, and volume-growth data to tie together conformal type, growth, and Sobolev-type inequalities, extending Kanai–Pittet–Bonk–Heinonen–Koskela-type results from Riemannian manifolds to a broad class of Lie groups and metric spaces with uniformly locally bounded geometry. These results yield rigidity and structure theorems for large-scale geometry and inform the interplay between infinitesimal conformality and global geometry in non-Riemannian settings.

Abstract

We study conditions under which quasi-conformal homeomorphisms are quasi-isometries. We show that if two nilpotent geodesic Lie groups are quasi-conformally homeomorphic, then they are quasi-isometrically equivalent. We also give more general results beyond the nilpotent case. In particular, we show that quasi-conformal homeomorphisms between geodesic Lie groups are quasi-isometries whenever the spaces have strict parabolic or hyperbolic conformal type. As a consequence, quasi-conformal homeomorphisms between geodesic Lie groups with infinite fundamental group are quasi-isometries. The statements for Lie groups are deduced from a more general study on metric measure spaces with uniformly locally bounded geometry.

Quasi-conformal VS quasi-isometric equivalence in spaces with controlled growth

TL;DR

This work establishes a comprehensive QC→QI principle for spaces with controlled growth, centering on geodesic Lie groups equipped with left-invariant geodesic distances and Haar measure. It develops a capacity–based framework using Ferrand distances in hyperbolic and parabolic conformal types to bridge infinitesimal quasiconformality with large-scale geometry, enabling bi-bornologous behavior and quasi-isometries under broad hypotheses. In particular, it proves that any quasi-conformal map between simply connected nilpotent geodesic Lie groups is a quasi-isometry, and more generally that QC maps between geodesic Lie groups are quasi-isometries when the conformal type is hyperbolic or strictly parabolic; the liminal parabolic case presents counterexamples. The methodology integrates capacity estimates, isoperimetric inequalities at large scale, and volume-growth data to tie together conformal type, growth, and Sobolev-type inequalities, extending Kanai–Pittet–Bonk–Heinonen–Koskela-type results from Riemannian manifolds to a broad class of Lie groups and metric spaces with uniformly locally bounded geometry. These results yield rigidity and structure theorems for large-scale geometry and inform the interplay between infinitesimal conformality and global geometry in non-Riemannian settings.

Abstract

We study conditions under which quasi-conformal homeomorphisms are quasi-isometries. We show that if two nilpotent geodesic Lie groups are quasi-conformally homeomorphic, then they are quasi-isometrically equivalent. We also give more general results beyond the nilpotent case. In particular, we show that quasi-conformal homeomorphisms between geodesic Lie groups are quasi-isometries whenever the spaces have strict parabolic or hyperbolic conformal type. As a consequence, quasi-conformal homeomorphisms between geodesic Lie groups with infinite fundamental group are quasi-isometries. The statements for Lie groups are deduced from a more general study on metric measure spaces with uniformly locally bounded geometry.
Paper Structure (41 sections, 63 theorems, 211 equations)

This paper contains 41 sections, 63 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Conformal type, volume growth, and isoperimetric inequalities
  5. QC maps in the hyperbolic conformal type
  6. QC maps in the parabolic conformal type
  7. Acknowledgments
  8. Spaces and maps
  9. Monotone functions
  10. Topological notions
  11. Monotone functions on topological spaces
  12. Estimate of the oscillation for monotone functions
  13. Lebesgue Straightening Lemma
  14. Conformal type
  15. Parabolicity and hyperbolicity
  16. ...and 26 more sections

Key Result

Theorem A

Let $G, H$ be nilpotent geodesic Lie groups. If there is a metrically quasi-conformal map $G\to H$, then there is a quasi-isometry $G\to H$.

Theorems & Definitions (143)

  • Theorem A: see Theorem \ref{['thm67f935d5']}
  • Theorem B: see Theorem \ref{['thm68780db5']}
  • Theorem C: see Theorem \ref{['thm68d55353']}
  • Theorem D: see Theorem \ref{['thm68655128']}
  • Theorem E: see Corollary \ref{['cor67f056ce']}
  • Theorem F: see Theorem \ref{['thm685cec65']}
  • Theorem G: see Theorem \ref{['thm68755ab4']}
  • Definition 2.1: Uniformly locally bounded geometry
  • Definition 2.2
  • Proposition 2.3
  • ...and 133 more