Quasi-Total Orders and Translation Numbers

TL;DR

This paper establishes a deep connection between non-zero homogeneous quasimorphisms and actions on posets, providing new insights and examples, including applications to Hermitian Lie groups and circle actions.

Contribution

It introduces a novel correspondence between quasimorphisms and poset actions, generalizing classical translation number constructions and applying to various important groups.

Findings

Characterizes when groups admit non-zero homogeneous quasimorphisms.

Provides new realizations of known quasimorphisms like Rademacher and Brooks types.

Links Guichardet-Wigner quasimorphisms to causal structures on Shilov boundaries.

Abstract

We show that a group admits a non-zero homogeneous quasimorphism if and only if it admits a certain type of action on a poset. Our proof is based on a construction of quasimorphisms which generalizes Poincar\'e--Ghys' construction of the classical translation number quasimorphism. We then develop a correspondence between quasimorphisms and actions on posets, which allows us to translate properties of orders into properties of quasimorphisms and vice versa. Concerning examples we obtain new realizations of the Rademacher quasimorphism, certain Brooks type quasimorphisms, the Dehornoy floor quasimorphism as well as Guichardet-Wigner quasimorphisms on simple Hermitian Lie groups of tube type. The latter we relate to Kaneyuki causal structures on Shilov boundaries, following an idea by Clerc and Koufany. As applications we characterize those quasimorphisms which arise from circle actions, and subgroups of Hermitian Lie groups with vanishing Guichardet-Wigner quasimorphisms.