On finite simple groups acting on homology spheres

Guazzi, Alessandra; Zimmermann, Bruno

doi:10.1007/s00605-012-0382-0

Cited by 7 publications

(4 citation statements)

References 14 publications

(17 reference statements)

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“…Let

p

be now an odd prime, let

d

be a positive integer, and consider a morphism

ψ : {normalΓ}_{d} \to Aut false(Γ_{p} false)

. The following is a very slight modification of a result of Guazzi and Zimmermann [, Lemma 2]: Lemma If

{normalΓ}_{p} ⋊_{ψ} {normalΓ}_{d}

acts effectively on a smooth

F_{p}

‐homology

n

‐sphere

S

and the restriction of the action to

Γ_{p}

is free then all elements in

ψ ({normalΓ}_{d})

have order dividing

n + 1

.…”

Section: Actions On Integral Homology Spheresmentioning

confidence: 99%

“…The original result of Guazzi and Zimmermann does not require the restriction of the action to

Γ_{p}

to be free, but it requires the action of

{normalΓ}_{p} ⋊_{ψ} {normalΓ}_{d}

to be orientation preserving. The proof we give of Lemma is essentially the same as [, Lemma 2]; we provide details to justify that the result is valid without assuming that

{normalΓ}_{p} ⋊_{ψ} {normalΓ}_{d}

acts orientation‐preservingly.…”

Section: Actions On Integral Homology Spheresmentioning

confidence: 99%

See 1 more Smart Citation

Finite group actions on homology spheres and manifolds with nonzero Euler characteristic

Riera

2019

Journal of Topology

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Let X be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology spheres. We prove that Diff(X) is Jordan. This means that there exists a constant C such that any finite subgroup G of Diff(X) has an abelian subgroup whose index in G is at most C. Using a result of Randall and Petrie, we deduce that the automorphism groups of connected, non-necessarily compact, smooth real affine varieties with nonzero Euler characteristic are Jordan.The proof of Theorem 1.2 uses a result by Turull and the author [36], which relies on the classification of finite simple groups (see Section 3).In order to explain the context of the results in this paper, it will be convenient to use the following terminology, introduced by Popov [39]: a group Γ is said to be Jordan if any finite subgroup G Γ has an abelian subgroup whose index in G is bounded above by a constant depending only on Γ.Theorem 1.2 gives a positive partial answer to the question, posed byÉtienne Ghys, of whether diffeomorphism groups of compact manifolds are Jordan (see Question 13.1 in [18] † ). The particular case in which X is a sphere was also independently asked in several talks by Walter Feit and later by Zimmermann [52, § 5].Other partial answers to Ghys's question have been given in the past. It has been proved that Diff(X) is Jordan if X is a compact manifold of dimension at most 2 (this is an easy exercise; see [30, Theorem 1.3] for the case of surfaces), a compact 3-manifold (Zimmermann [53]), a compact 4-manifold with nonzero Euler characteristic [31], or a closed manifold with a nonzero top dimensional cohomology class expressible as a product of one-dimensional classes (for example, tori) [30]. One can also study Jordan's property for diffeomorphism groups of open manifolds. It is easy to prove that Diff(R) and Diff(R 2 ) are Jordan; the work of Meeks and Yau on minimal surfaces [27, Theorem 4] implies that Diff(R 3 ) is Jordan; Guazzi, Mecchia and Zimmermann later proved in [20] that Diff(R 3 ) and Diff(R 4 ) are Jordan using much more elementary methods than those of [27], and Zimmermann proved in [53] that Diff(R 5 ) and Diff(R 6 ) are Jordan.An example of connected open 4-manifold whose diffeomorphism group is not Jordan was given by Popov in [40]. Csikós, Pyber, and Szabó [11] found the first example giving a negative answer to Ghys's question, showing that the diffeomorphism group of T 2 × S 2 is not Jordan (but note that, in contrast, the symplectomorphism group of any symplectic form on T 2 × S 2 is Jordan; see [33]). Many other examples of manifolds giving a negative answer to Ghys's question can be obtained using the ideas in [11]: in particular, for any manifold M supporting an effective action of SU(2) or SO(3, R), the diffeomorphism group of T 2 × M is not Jordan (see [32]). It is an intriguing question to characterize which compact smooth manifolds have Jordan diffeomorph...

show abstract

“…Let

p

be now an odd prime, let

d

be a positive integer, and consider a morphism

ψ : {normalΓ}_{d} \to Aut false(Γ_{p} false)

. The following is a very slight modification of a result of Guazzi and Zimmermann [, Lemma 2]: Lemma If

{normalΓ}_{p} ⋊_{ψ} {normalΓ}_{d}

acts effectively on a smooth

F_{p}

‐homology

n

‐sphere

S

and the restriction of the action to

Γ_{p}

is free then all elements in

ψ ({normalΓ}_{d})

have order dividing

n + 1

.…”

Section: Actions On Integral Homology Spheresmentioning

confidence: 99%

“…The original result of Guazzi and Zimmermann does not require the restriction of the action to

Γ_{p}

to be free, but it requires the action of

{normalΓ}_{p} ⋊_{ψ} {normalΓ}_{d}

to be orientation preserving. The proof we give of Lemma is essentially the same as [, Lemma 2]; we provide details to justify that the result is valid without assuming that

{normalΓ}_{p} ⋊_{ψ} {normalΓ}_{d}

acts orientation‐preservingly.…”

Section: Actions On Integral Homology Spheresmentioning

confidence: 99%

Finite group actions on homology spheres and manifolds with nonzero Euler characteristic

Riera

2019

Journal of Topology

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“…For the proof of Theorem 2 we have to exclude all of these groups except A 5 ∼ = PSL 2 (5), PSL 2 (7) and PSU 3 (q). This is based on the following result from [10] (resp. on some part of its proof): Theorem 4.…”

Section: Proofsmentioning

confidence: 99%

On finite simple groups acting on homology spheres with small fixed point sets

Zimmermann

2014

Bol. Soc. Mat. Mex.

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A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets ("pseudofree action") is the alternating group A 5 acting on the 2-sphere. Our first main theorem is the finiteness result that there are only finitely many finite simple groups which admit a smooth action on a homology sphere with at most d-dimensional fixed points sets, for a fixed d. We then go on proving that the finite simple groups acting on a homology sphere with at most 1-dimensional fixed point sets are the alternating group A 5 in dimensions 2, 3 and 5, the linear fractional group PSL 2 (7) in dimension 5, and possibly the unitary group PSU 3 (3) in dimension 5 (we conjecture that it does not admit any action on a homology 5-sphere but cannot exclude it at present). Finally, we discuss the situation for arbitrary finite groups which admit an action on a homology 3-sphere.

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“…Recently B. Csikós, L. Pyber, and E. Szabó in [CPS14] provided a counterexample following the method of [Zar14]; see also [Mun17b] for a further development of this method, and [Pop16, Corollary 2] for a non-compact counterexample. However, Jordan property holds for diffeomorphism groups in many cases; see [Mun16], [Mun14], [MT15], [Mun13], [GZ13], [Zim12], [Zim14a], [Zim14b], [MZ15], and references therein. Also there are results for groups of symplectomorphisms, see [Mun17a] and [Mun18].…”

Section: Introductionmentioning

confidence: 99%

Automorphism Groups of Compact Complex Surfaces

Prokhorov

Shramov

2019

International Mathematics Research Notices

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We classify compact complex surfaces whose groups of bimeromorphic selfmaps have bounded finite subgroups. We also prove that the stabilizer of a point in the automorphism group of a compact complex surface of zero Kodaira dimension, as well as the stabilizer of a point in the automorphism group of an arbitrary compact Kähler manifold of non-negative Kodaira dimension, always has bounded finite subgroups.

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On finite simple groups acting on homology spheres

Cited by 7 publications

References 14 publications

Finite group actions on homology spheres and manifolds with nonzero Euler characteristic

Finite group actions on homology spheres and manifolds with nonzero Euler characteristic

On finite simple groups acting on homology spheres with small fixed point sets

Automorphism Groups of Compact Complex Surfaces

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