[Paper Review] Sublinear quasiconformality and the large-scale geometry of Heintze groups

Gabriel Pallier|arXiv (Cornell University)|May 22, 2019

Geometric Analysis and Curvature Flows58 references4 citations

TL;DR

This paper introduces sublinear quasiconformality as a generalization of quasisymmetry to study the large-scale geometry of Heintze groups and negatively curved spaces. It defines a new conformal dimension invariant, $\operatorname{Cdim}_{O(u)}$, which distinguishes certain Riemannian negatively curved homogeneous spaces and Fuchsian buildings up to sublinear biLipschitz equivalence, and proves that $O(u)$-sublinearly biLipschitz equivalent purely real Heintze groups with abelian nilradicals have isomorphic $S_\infty$-limits, resolving a conjecture for dimension 3.

ABSTRACT

This article analyzes sublinearly quasisymmetric homeo-morphisms (generalized quasisymmetric mappings), and draws applications to the sublinear large-scale geometry of negatively curved groups and spaces. It is proven that those homeomorphisms lack analytical properties but preserve a conformal dimension and appropriate function spaces, distinguishing certain (nonsymmetric) Riemannian negatively curved homogeneous spaces, and Fuchsian buildings, up to sublinearly biLipschitz equivalence (generalized quasiisometry).

Motivation & Objective

• To extend the theory of quasisymmetric mappings to sublinearly quasiconformal homeomorphisms for studying large-scale geometry.
• To define and compute a new conformal dimension invariant $\operatorname{Cdim}_{O(u)}$ that is preserved under sublinear biLipschitz equivalences.
• To classify purely real Heintze groups with abelian nilradicals up to sublinear biLipschitz equivalence using the $S_\infty$-limit construction.
• To show that this invariant distinguishes Fuchsian buildings and certain Riemannian negatively curved spaces up to sublinear biLipschitz equivalence.

Proposed method

• Introduces $O(u)$-quasisymmetric homeomorphisms using an asymptotic class $O(u)$ with admissible sublinear functions $u$, generalizing quasisymmetry.
• Defines a new conformal dimension $\operatorname{Cdim}_{O(u)}$ as a coarse invariant of the Gromov boundary of hyperbolic spaces.
• Constructs function spaces of locally bounded $p$-variation invariant under sublinear quasisymmetric maps up to parameter shifts.
• Uses the $S_\infty$-limit construction to relate sublinear biLipschitz equivalence to isomorphism of asymptotic structures in Heintze groups.
• Applies the theory to compute $\operatorname{Cdim}_{O(u)}$ for Bourdon buildings, showing it equals the classical conformal dimension.
• Employs metric estimates and logarithmic distortion control via admissible functions $u$ to prove invariance and stability of the new conformal dimension.

Experimental results

Research questions

• RQ1 Does the $O(u)$-sublinear biLipschitz equivalence of purely real Heintze groups imply isomorphism of their $S_\infty$-limits?
• RQ2 Can the new conformal dimension $\operatorname{Cdim}_{O(u)}$ distinguish non-isometric Fuchsian buildings up to sublinear biLipschitz equivalence?
• RQ3 Do sublinear quasisymmetric maps preserve function spaces of bounded $p$-variation up to parameter shifts?
• RQ4 Is $\operatorname{Cdim}_{O(u)}$ equal to the classical Pansu conformal dimension on Heintze groups and Bourdon buildings?
• RQ5 Does the failure of ACL property in $O(u)$-quasisymmetric maps necessitate a shift from analytical to global geometric invariants?

Key findings

• For purely real Heintze groups with abelian nilradicals, $O(u)$-sublinear biLipschitz equivalence implies isomorphism of their $S_\infty$-limits, confirming a conjecture by Cornulier for dimension 3.
• The conformal dimension $\operatorname{Cdim}_{O(u)}$ of the boundary of a Bourdon building $I_{pq}$ equals $1 + \frac{\log(q-1)}{\log(p-1)}$, matching the classical conformal dimension.
• The invariant $\operatorname{Cdim}_{O(u)}$ is preserved under sublinear quasisymmetric homeomorphisms and distinguishes certain Riemannian negatively curved homogeneous spaces.
• Function spaces of bounded $p$-variation are preserved under sublinear quasisymmetric maps up to shifts in the parameter $p$, providing additional invariants.
• The $O(u)$-quasisymmetric structure on $\mathbf{R}^2$ is well-defined and stable under product maps, even though the ACL property fails.
• The new conformal dimension $\operatorname{Cdim}_{O(u)}$ coincides with the classical $\operatorname{Cdim}$ on examples studied by Pansu and Bourdon.