The identity component of the leaf preserving diffeomorphism group is perfect

Rybicki, Tomasz

doi:10.1007/bf01294862

Cited by 29 publications

(23 citation statements)

References 17 publications

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“…, k or r = ∞. For example, for homeomorphisms (class C 0 ) this theorem was proven by Fukui and Imanishi in [5], and for the class C ∞ by Rybicki [11]. It is easy to check that the assumptions of Theorem 1.6 are fulfilled.…”

Section: Examplesmentioning

confidence: 82%

Quasi-simplicity of some homeomorphism and diffeomorphism groups

Kowalik¹,

Michalik²

2010

Demonstratio Mathematica

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Abstract. The notion of quasi-simplicity of groups is introduced. It is proven that for a group of homeomorphisms G which is fixed point free and factorizable the commutator subgroup [G, G] is quasi-simple. Several examples of quasi-simple but non-simple homeomorphism groups are presented. IntroductionIn paper [8] Ling showed that for a factorizable and fixed point free group G the commutator subgroup [G, G] is the least subgroup of G, normalized by [G, G], which acts without any fixed point. In this note we introduce the notion of a quasi-simple group. This notion enables us to give a new formulation of Ling's theorem and a completely new proof of it. In the last section we present various examples of quasi-simple homeomorphism and diffeomorphism groups which are non-simple.Ling's theorem is a generalization of the following theorem of Epstein [4]: Theorem 1.1. Let X be a paracompact space, let G be a group of homeomorphisms of X and let U be a basis of open sets of X satisfying the following axioms:Axiom 3. Let g ∈ G, U ∈ U and let B ⊆ U be a covering of X. Then there exist an integer n, elements g 1 , . . . , g n ∈ G and V 1 , . . . , V n ∈ B such that g = g n g n−1 . . . g 1 , supp(g i ) ⊂ V i and supp(g i ) ∪ (g i−1 . . . g 1 (U )) = X for 1 i n.Then [G, G], the commutator subgroup of G, is simple.2000 Mathematics Subject Classification: 22A05, 22E65, 57S05.

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Section: Examplesmentioning

confidence: 82%

Quasi-simplicity of some homeomorphism and diffeomorphism groups

Kowalik¹,

Michalik²

2010

Demonstratio Mathematica

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“…T.Tsuboi [12] (and T.Rybicki [7]) proved the following by looking at the proofs in Herman [6] and Thurston [11]. …”

Section: Corollary 25mentioning

confidence: 99%

On the First Homology of the Groups of Foliation Preserving Diﬀeomorphisms for Foliations with Singularities of Morse Type

Fukui¹

2008

Publ. Res. Inst. Math. Sci.

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Let R n be an n-dimensional Euclidean space and F ϕ be the foliation defined by levels of a Morse function ϕ : R n → R. We determine the first homology of the identity component of the foliation preserving diffeomorphism group of (R n , F ϕ ). Then we can apply it to the calculation of the first homology of the foliation preserving diffeomorphism groups for codimension one compact foliations with singularities of Morse type.

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“…The more general reason is that the Alexander trick is no longer true, and consequently the definition of homology becomes much more complicated in that case (cf. [1], [6]). However it is still true that H\(Q°°(n,k)0) = 0, where Q°°(n, k) stands for the group of leaf preserving diffeomorphisms of class C°° on (R n , F^), but the proof is much longer and difficult [6].…”

Section: G(xy)mentioning

confidence: 99%

“…[1], [6]). However it is still true that H\(Q°°(n,k)0) = 0, where Q°°(n, k) stands for the group of leaf preserving diffeomorphisms of class C°° on (R n , F^), but the proof is much longer and difficult [6].…”

Section: G(xy)mentioning

confidence: 99%