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Coherent actions by homeomorphisms on the real line or an interval
Group Theory · Submitted 26 Feb 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We establish that such an action is rigid, i.e. any two such actions of the same group are topologically conjugate.
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The abstract, sentence by sentence
We study actions of groups by homeomorphisms on $\mathbf{R}$ (or an interval) that are minimal, have solvable germs at $\pm \infty$ and contain a pair of elements of a certain type. We call such actions coherent. We establish that such an action is rigid, i.e. any two such actions of the same group are topologically conjugate. We also establish that the underlying group is always non elementary amenable, but satisfies that every proper quotient is solvable. As a first application, we demonstrate that any coherent group action $G<Homeo^+(\mathbf{R})$ that produces a nonamenable equivalence relation with respect to the Lebesgue measure satisfies that the underlying group does not embed into Thompson's group $F$. This includes all known examples of nonamenable groups that do not contain non abelian free subgroups and act faithfully on the real line by homeomorphisms. As a second application, we establish that the Brown-Stein-Thompson groups $F(2,p_1,...,p_n)$ for $n\geq 1$ and $p_1,...,p_n$ distinct odd primes, do not embed into Thompson's group $F$. This answers a question recently raised by C. Bleak, M. Brin, and J. Moore. Our tools also allow us to prove additional non embeddability results for Brown-Stein-Thompson and Bieri-Strebel groups.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study actions of groups by homeomorphisms on $\mathbf{R}$ (or an interval) that are minimal, have solvable germs at $\pm \infty$ and contain a pair of elements of a certain type. | contribution 0.91 | 0.65 | 0.04 | asserted 0.67 |
| s1 | We call such actions coherent. | contribution 0.59 | 0.66 | 0.03 | asserted 0.85 |
| s2 | We establish that such an action is rigid, i.e. any two such actions of the same group are topologically conjugate. | result 1.00 | 0.96 | 0.69 | asserted 1.00 |
| s3 | We also establish that the underlying group is always non elementary amenable, but satisfies that every proper quotient is solvable. | result 1.00 | 0.94 | 0.18 | asserted 1.00 |
| s4 | As a first application, we demonstrate that any coherent group action $G<Homeo^+(\mathbf{R})$ that produces a nonamenable equivalence relation with respect to the Lebesgue measure satisfies that the underlying group does not embed into Thompson's group $F$. | result 0.89 | 0.95 | 0.14 | asserted 1.00 |
| s5 | This includes all known examples of nonamenable groups that do not contain non abelian free subgroups and act faithfully on the real line by homeomorphisms. | interpretation 0.74 | 0.54 | 0.07 | asserted 0.99 |
| s6 | As a second application, we establish that the Brown-Stein-Thompson groups $F(2,p_1,...,p_n)$ for $n\geq 1$ and $p_1,...,p_n$ distinct odd primes, do not embed into Thompson's group $F$. | result 0.96 | 0.95 | 0.10 | asserted 1.00 |
| s7 | This answers a question recently raised by C. Bleak, M. Brin, and J. Moore. | interpretation 0.37 | 0.43 | 0.05 | asserted 0.98 |
| s8 | Our tools also allow us to prove additional non embeddability results for Brown-Stein-Thompson and Bieri-Strebel groups. | result 0.69 | 0.94 | 0.05 | asserted 0.98 |
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Original abstract
We study actions of groups by homeomorphisms on $\mathbf{R}$ (or an interval) that are minimal, have solvable germs at $\pm \infty$ and contain a pair of elements of a certain type. We call such actions coherent. We establish that such an action is rigid, i.e. any two such actions of the same group are topologically conjugate. We also establish that the underlying group is always non elementary amenable, but satisfies that every proper quotient is solvable. As a first application, we demonstrate that any coherent group action $G<Homeo^+(\mathbf{R})$ that produces a nonamenable equivalence relation with respect to the Lebesgue measure satisfies that the underlying group does not embed into Thompson's group $F$. This includes all known examples of nonamenable groups that do not contain non abelian free subgroups and act faithfully on the real line by homeomorphisms. As a second application, we establish that the Brown-Stein-Thompson groups $F(2,p_1,...,p_n)$ for $n\geq 1$ and $p_1,...,p_n$ distinct odd primes, do not embed into Thompson's group $F$. This answers a question recently raised by C. Bleak, M. Brin, and J. Moore. Our tools also allow us to prove additional non embeddability results for Brown-Stein-Thompson and Bieri-Strebel groups.
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What it cites · 3 sentences without a confident label
Cites as background 7 papers
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Such groups were studied extensively in [ref] .
Chain groups of homeomorphisms of the interval · Examples of coherent actions · 0.99
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Thompson more than $50$ years ago, and the group is an interesting and important infinite group that has been postulated as a counterexample for various conjectures in group theory (see [ref] for instance.) The subgroup structure of $F$ is quite mysterious and several recent papers (for instance [ref] , [ref] , [ref] and [ref] ) have been devoted to developing a systematic structure theory of subgroups of $F$.
On subgroups of R. Thompson's group $F$ · Introduction · 1.00
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Thompson more than $50$ years ago, and the group is an interesting and important infinite group that has been postulated as a counterexample for various conjectures in group theory (see [ref] for instance.) The subgroup structure of $F$ is quite mysterious and several recent papers (for instance [ref] , [ref] , [ref] and [ref] ) have been devoted to developing a systematic structure theory of subgroups of $F$.
The generation problem in Thompson group $F$ · Introduction · 1.00
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