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A Tukia-type theorem for nilpotent Lie groups and quasi-isometric rigidity of solvable groups

Tullia Dymarz, David Fisher, Xiangdong Xie

Abstract

In this paper we study uniform quasiconformal groups of Carnot-by-Carnot groups. We show that they can be conjugated into conformal groups provided the induced action on the space of distinct pairs is cocompact. Following the approach of Eskin-Fisher-Whyte these results have applications to quasi-isometric rigidity of certain solvable groups.

A Tukia-type theorem for nilpotent Lie groups and quasi-isometric rigidity of solvable groups

Abstract

In this paper we study uniform quasiconformal groups of Carnot-by-Carnot groups. We show that they can be conjugated into conformal groups provided the induced action on the space of distinct pairs is cocompact. Following the approach of Eskin-Fisher-Whyte these results have applications to quasi-isometric rigidity of certain solvable groups.
Paper Structure (31 sections, 55 theorems, 148 equations)

This paper contains 31 sections, 55 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Outline of main theorem
  3. Preliminary
  4. Quasi-actions and uniform quasisimilarity actions
  5. Nilpotent Lie algebras and nilpotent Lie groups
  6. Carnot algebras and Carnot groups
  7. Derivations, semi-direct products, Heintze groups, and SOL-like groups
  8. Homogeneous distances on nilpotent Lie groups
  9. A fiber Tukia theorem for diagonal Heintze pairs
  10. Quasi-isometric rigidity of solvable groups
  11. Quasi-isometric rigidity of lattices in the isometry group of SOL-like groups
  12. Quasi-isometric classification of a class of solvable Lie groups
  13. Quasi-isometric rigidity of quasi-actions on certain Heintze groups
  14. BiLipschitz maps of diagonal Heintze pairs
  15. Characterization of biLipschitz shear maps
  16. ...and 16 more sections

Key Result

Theorem 1.2

Let $(N,D)$ be a Carnot-by-Carnot group and $\Gamma$ a group with a uniform quasisimilarity action of $N$ such that almost every point in $N$ is a radial limit point. When $\dim(\mathfrak w)=1$, we further assume $\Gamma$ is locally compact amenable. Then there exists a biLipschitz map $F_0$ of $N$

Theorems & Definitions (108)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Conjecture 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 3.1
  • ...and 98 more