On Lipschitz functions on groups equipped with conjugation-invariant norms

TL;DR

This paper characterizes Lipschitz functions on groups with conjugation-invariant norms as partial quasimorphisms and explores their properties, including homogenization and detection of undistorted elements.

Contribution

It establishes a precise equivalence between Lipschitz functions and partial quasimorphisms on such groups, and introduces a homogenization method linking these functions to asymptotic cones.

Findings

Lipschitz functions correspond to bounded partial quasimorphisms

Undistorted elements are detected by antisymmetric homogeneous partial quasimorphisms

A homogenization procedure relates partial quasimorphisms to asymptotic cones

Abstract

We observe that a function on a group equipped with a bi-invariant word metric is Lipschitz if and only if it is a partial quasimorphism bounded on the generating set. We also show that an undistorted element is always detected by an antisymmetric homogeneous partial quasimorphisms. We provide a general homogenisation procedure for Lipschitz functions and relate partial quasimorphisms on a group to ones on its asymptotic cones.