# Sublinear quasiconformality and the large-scale geometry of Heintze groups

## TL;DR

This paper studies sublinear quasisymmetric homeomorphisms and their impact on the large-scale geometry of negatively curved groups and spaces, revealing their properties and classification capabilities.

## Contribution

It introduces the analysis of sublinear quasisymmetric mappings and demonstrates their role in classifying negatively curved spaces up to sublinear biLipschitz equivalence.

## Key findings

- Homeomorphisms preserve conformal dimension and function spaces.
- They distinguish certain negatively curved spaces and Fuchsian buildings.
- Analytical properties of these homeomorphisms are limited.

## 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).