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Quasi-Total Orders and Translation Numbers
Group Theory · Submitted 30 Jun 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | result 0.98 | 0.95 | 0.90 | asserted 1.00 |
| s1 | Our proof is based on a construction of quasimorphisms which generalizes Poincar\'e--Ghys' construction of the classical translation number quasimorphism. | method 0.96 | 0.93 | 0.05 | asserted 1.00 |
| s2 | 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. | contribution 0.69 | 0.95 | 0.13 | asserted 1.00 |
| s3 | 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. | result 0.62 | 0.93 | 0.04 | asserted 1.00 |
| s4 | The latter we relate to Kaneyuki causal structures on Shilov boundaries, following an idea by Clerc and Koufany. | contribution 0.51 | 0.73 | 0.03 | asserted 0.99 |
| s5 | As applications we characterize those quasimorphisms which arise from circle actions, and subgroups of Hermitian Lie groups with vanishing Guichardet-Wigner quasimorphisms. | result 0.82 | 0.92 | 0.09 | asserted 1.00 |
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The paper
Original 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.
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How the literature uses it
3 arXiv papers cite this in 11 sentences
Papers citing it · 6 sentences without a confident label
Builds on it 1 paper
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We revisit in the special case of the Shilov boundary the theory developed for general causal covers in [ref] and provide a new self-contained proof of the following result (see [ref] ) establishing the relation between the order on $G$ and $f_G$.
On weakly maximal representations of surface groups · Orders and Bounded Cohomology · 0.83
Cites it as support 1 paper
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This is established implicitly in [ref] and more explicitly in [ref] .
Invariant Orders on Hermitian Lie Groups · The geometric order and the maximal order · 1.00
Cites it as background 2 papers
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(It is easy to see that $G$ acts effectively on $Ř$ so that this is well-defined.) The geometric order was first explored systematically in [ref] , see also [ref] and [ref] .
Invariant Orders on Hermitian Lie Groups · Introduction · 0.91
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The reader is also referred to Ben Simon and Hartnick's work [ref] regarding a general connection between quasi-morphisms and partial orders.
Quasi-morphisms on contactomorphism groups and contact rigidity · Orderability for contact manifolds and quasi-morphisms · 0.99
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