Groups, Geometry, and Dynamics

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Volume 9, Issue 2, 2015, pp. 479–530
DOI: 10.4171/GGD/319

Published online: 2015-06-08

Quasi-total actions and translation numbers

Gabi Ben-Simon[1] and Tobias Hartnick[2]

(1) ORT Braude College, Karmiel, Israel
(2) Technion - Israel Institute of Technology, Haifa, Israel

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 the 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.

Keywords: Quasimorphism, partially ordered groups, bounded cohomology, Hermitian Lie groups

Ben-Simon Gabi, Hartnick Tobias: Quasi-total actions and translation numbers. Groups Geom. Dyn. 9 (2015), 479-530. doi: 10.4171/GGD/319