[Paper Review] Rigidity of fiber-preserving quasisymmetric maps

Enrico Le Donne, Xiangdong Xie|arXiv (Cornell University)|Jan 10, 2015

Geometric Analysis and Curvature Flows9 references3 citations

TL;DR

This paper establishes that fiber-preserving quasisymmetric maps between certain metric spaces—specifically, $(\alpha,L)$-fibered quasimetric spaces—are necessarily biLipschitz. The key result shows that under conditions of parallelism, non-isolation, and divergence of non-parallel fibers, quasisymmetric maps preserving fibers must be quasi-similarities, hence biLipschitz. This rigidity result is applied to Carnot groups with reducible first stratum and to Heisenberg groups, proving that quasisymmetric maps preserving vertical lines are biLipschitz.

ABSTRACT

We show that fiber-preserving quasisymmetric maps are biLipschitz. As an application, we show that quasisymmetric maps on Carnot groups with reducible first stratum are biLipschitz.

Motivation & Objective

• To establish a general rigidity condition under which quasisymmetric maps preserving a foliation (fibers) are biLipschitz.
• To define and analyze the structure of $(\alpha,L)$-fibered quasimetric spaces, capturing key geometric features of ideal boundaries of negatively curved solvable Lie groups.
• To apply the main rigidity result to quasisymmetric maps on Carnot groups with reducible first stratum, proving they are biLipschitz.
• To extend the result to quasiconformal maps on Heisenberg groups that preserve vertical lines, showing they are also biLipschitz.
• To provide a new proof of Dymarz's theorem on quasisymmetric maps of $N \times \mathbb{Q}_m$ being biLipschitz.

Proposed method

• Introduce the concept of an $(\alpha,L)$-fibered quasimetric space, requiring fibers to be snowflake-equivalent to unbounded geodesic spaces and satisfying parallelism and divergence conditions.
• Define fiber-preserving quasisymmetric maps and show they are quasi-similarities—hence biLipschitz—by controlling distortion using quasimetric constants and the parameters $\alpha$, $L$, and $\eta$.
• Use the structure of Carnot groups, particularly the stratification of their Lie algebras and the action of strata-preserving automorphisms, to verify the fibered quasimetric space axioms.
• Apply the main theorem to the Heisenberg group $H^n$, where fibers are vertical lines, showing that quasisymmetric maps preserving them are biLipschitz via the $\alpha = 1/2$, $L$-biLipschitz equivalence to $\mathbb{R}$ with the $1/2$-snowflake metric.
• Prove that for $N \times \mathbb{Q}_m$ with $N$ a Carnot group and $\mathbb{Q}_m$ the $m$-adic integers, all quasisymmetric maps are biLipschitz by verifying all axioms of the fibered quasimetric space and applying Theorem 1.1.
• Use the Baker–Campbell–Hausdorff formula and properties of Lie brackets to analyze the behavior of group elements under conjugation and dilation, leading to a contradiction when assuming finite Hausdorff distance between a subgroup and its conjugate unless the conjugating element normalizes the subgroup.

Experimental results

Research questions

• RQ1 Under what conditions on a quasimetric space does a fiber-preserving quasisymmetric map become biLipschitz?
• RQ2 Can the rigidity of quasisymmetric maps on ideal boundaries of negatively curved solvable Lie groups be reduced to a fiber-preserving structure?
• RQ3 Are quasisymmetric maps on Carnot groups with reducible first stratum necessarily biLipschitz?
• RQ4 Do quasiconformal maps on the Heisenberg group that preserve vertical lines have to be biLipschitz?
• RQ5 Is every quasisymmetric map on $N \times \mathbb{Q}_m$ for a Carnot group $N$ and $m$-adic integers $\mathbb{Q}_m$ necessarily biLipschitz?

Key findings

• Any $\eta$-quasisymmetric map $F: X \to Y$ between $(\alpha,L)$-fibered quasimetric spaces that preserves fibers is a $(K,C)$-quasi-similarity, hence biLipschitz, with $K$ depending only on $\eta$, $\alpha$, $L$, and the quasimetric constants of $X$ and $Y$.
• For the Heisenberg group $H^n$, any quasiconformal map preserving vertical lines is biLipschitz, as vertical lines are $L$-biLipschitz to $\mathbb{R}$ with the $1/2$-snowflake metric.
• Every quasisymmetric map on $N \times \mathbb{Q}_m$ equipped with the max metric is biLipschitz, as the fibers $N \times \{y\}$ satisfy all axioms of a fibered quasimetric space.
• The condition $d(u,V) = d(v,U) = a$ for parallel fibers cannot be weakened to $HD(U,V) < \infty$; a counterexample is provided to show the necessity of the stronger parallelism condition.
• In a Carnot group with reducible first stratum, the existence of a non-trivial $A$-invariant subspace in $V_1$ for all strata-preserving automorphisms $A$ leads to a contradiction if a conjugating element $g$ does not normalize the corresponding subgroup, proving that such maps must be biLipschitz.