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Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds
Differential Geometry · Submitted 22 Dec 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In particular we prove that for any $\epsilon>0$ there exist manifolds admitting effective actions of arbitrarily large $p$-groups $\Gamma$ all of whose abelian subgroups have at most $|\Gamma|^{\epsilon}$ elements.
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The abstract, sentence by sentence
A recent preprint of Csik\'os, Pyber and Szab\'o (arXiv:1411.7524) proves that the diffeomorphism group of $T^2\times S^2$ is not Jordan. The purpose of this paper is to generalize the arguments of Csik\'os, Pyber and Szab\'o in order to obtain many other examples of compact manifolds whose diffeomorphism group fails to be Jordan. In particular we prove that for any $\epsilon>0$ there exist manifolds admitting effective actions of arbitrarily large $p$-groups $\Gamma$ all of whose abelian subgroups have at most $|\Gamma|^{\epsilon}$ elements. Finally, we also recover some results on nonexistence of effective actions of compact connected semisimple Lie group on manifolds.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A recent preprint of Csik\'os, Pyber and Szab\'o (arXiv:1411.7524) proves that the diffeomorphism group of $T^2\times S^2$ is not Jordan. | background 1.00 | 0.03 | 0.05 | asserted 1.00 |
| s1 | The purpose of this paper is to generalize the arguments of Csik\'os, Pyber and Szab\'o in order to obtain many other examples of compact manifolds whose diffeomorphism group fails to be Jordan. | contribution 0.97 | 0.39 | 0.09 | asserted 0.90 |
| s2 | In particular we prove that for any $\epsilon>0$ there exist manifolds admitting effective actions of arbitrarily large $p$-groups $\Gamma$ all of whose abelian subgroups have at most $|\Gamma|^{\epsilon}$ elements. | result 0.98 | 0.93 | 0.55 | asserted 1.00 |
| s3 | Finally, we also recover some results on nonexistence of effective actions of compact connected semisimple Lie group on manifolds. | result 0.92 | 0.76 | 0.06 | asserted 0.99 |
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The paper
Original abstract
A recent preprint of Csik\'os, Pyber and Szab\'o (arXiv:1411.7524) proves that the diffeomorphism group of $T^2\times S^2$ is not Jordan. The purpose of this paper is to generalize the arguments of Csik\'os, Pyber and Szab\'o in order to obtain many other examples of compact manifolds whose diffeomorphism group fails to be Jordan. In particular we prove that for any $\epsilon>0$ there exist manifolds admitting effective actions of arbitrarily large $p$-groups $\Gamma$ all of whose abelian subgroups have at most $|\Gamma|^{\epsilon}$ elements. Finally, we also recover some results on nonexistence of effective actions of compact connected semisimple Lie group on manifolds.
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How the literature uses it
5 arXiv papers cite this in 10 sentences
Papers citing it · 6 sentences without a confident label
Builds on it 1 paper
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Many other counterexamples to Ghys's conjecture can be obtained using the ideas in [ref] : in particular, for any manifold $M$ supporting an effective action of $(2)$ or $(3,)$ the diffeomorphism group of $T^2×M$ is not Jordan (see [ref] ).
Finite group actions on homology spheres and manifolds with nonzero Euler characteristic · Section · 0.85
Cites it as support 1 paper
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Statement (3) in Theorem [ref] is also sharp: this follows from [ref] (see also [ref] ).
Which finite groups act smoothly on a given $4$-manifold? · Geometric structures · 0.99
Cites it as background 2 papers
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A number of papers have been written on this question in the last few years [ref] .
Which finite groups act smoothly on a given $4$-manifold? · Jordan property · 0.92
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A number of papers have recently been devoted to Ghys's question (see e.g. [ref] and the references therein), and it is known that the answer is affirmative in many cases and negative in many other ones [ref] , although a description of which compact smooth manifolds have Jordan diffeomorphism group seems to be at present a widely open problem.
Finite subgroups of Ham and Symp · Section · 0.92
What it cites · 6 sentences without a confident label
Compares with 2 papers
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The main result in [ref] implies that $Λ((T^2×S^2))≤2/3$, while our theorem implies that if $X$ supports an almost effective action of $(τ(n))$ then $Λ((T^2n×X))≤(n+1)/(2n+1)$ for any $n$.
Diffeomorphism Groups of Compact 4-manifolds are not always Jordan · Introduction · 0.87
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Previously, Popov [ref] had given an example of a connected open $4$-manifold with non Jordan diffeomorphism group.
Finite subgroups of diffeomorphism groups · Introduction · 0.87
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