On Lipschitz functions on groups equipped with conjugation-invariant norms

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 a generating set. We also show that an undistorted element is always detected by an antisymmetric homogeneous partial quasimorphism. We provide a general homogenisation procedure for Lipschitz functions and relate partial quasimorphisms on a group to ones on its asymptotic cones.