Rigidity of certain solvable actions on the sphere

Asaoka, Masayuki

doi:10.2140/gt.2012.16.1835

Cited by 11 publications

(32 citation statements)

References 19 publications

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“…Another way of seeing this is that C 1 continuity follows from the choice of P Z for the set of breakpoints 3. To see this, first check that the relations of F are satisfied and conclude using the property F satisfies that every proper quotient is abelian.…”

mentioning

confidence: 99%

Hyperbolicity as an obstruction to smoothability for one-dimensional actions

2019

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Ghys and Sergiescu proved in the 80s that Thompson's group T , and hence F , admits actions by C ∞ diffeomorphisms of the circle . They proved that the standard actions of these groups are topologically conjugate to a group of C ∞ diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha-Moore defined finitely presentable groups of piecewise projective homeomorphisms. These groups are of particular interest because they are nonamenable and contain no free subgroup. In contrast to the result of Ghys-Sergiescu, we prove that the groups of Monod and Lodha-Moore are not topologically conjugate to a group of C 1 diffeomorphisms.Furthermore, we show that the group of Lodha-Moore has no nonabelian C 1 action on the interval. We also show that many Monod's groups H(A), for instance when A is such that PSL(2, A) contains a rational homothety x → p q x, do not admit a C 1 action on the interval. The obstruction comes from the existence of hyperbolic fixed points for C 1 actions. With slightly different techniques, we also show that some groups of piecewise affine homeomorphisms of the interval or the circle are not smoothable. 1 2 Some definitions and notation Definition 2.1. Let M be a manifold and Homeo(M ) the group of homeomorphisms of M . A subgroup G ⊂ Homeo(M ) is C r -smoothable (r ≥ 1) if it is conjugate in Homeo(M ) to a subgroup in Diff r (M ), the group of C r diffeomorphisms of M .Remark 2.2. Even if a certain subgroup G ⊂ Homeo(M ) is not C r -smoothable, it is still possible that the group G, as abstract group, admits C r actions on the manifold M .

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confidence: 99%

Hyperbolicity as an obstruction to smoothability for one-dimensional actions

2019

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“…Combined with Proposition 2.2, the theorem implies Theorem 1.5. The above theorem follows from the same argument as in [1]. Firstly, we prove the stability of the linear part of the local action.…”

Section: Rigidity Of Local Actionsmentioning

confidence: 88%

“…The theorem is proved by an application of the method used in [1]. Firstly, we show persistence of the global fixed point x ∞ .…”

Section: Theorem 11 (Burslem and Wilkinsonmentioning

confidence: 90%

“…The sphere S n admits a natural conformal structure and the action ρ B preserves it. In [1], the author of this paper proved that the actionρ B is not locally rigid but it exhibits a weak form of rigidity.…”

Section: Theorem 11 (Burslem and Wilkinsonmentioning

confidence: 98%

“…For integers n ≥ 1 and k ≥ 2, let Γ n,k be the finitely presented group given by where we set c • ∞ = ∞ and ∞ + t = ∞ for any c ̸ = 0 and t ∈ R. This action preserves the standard projective structure on RP 1 . In [2], Burslem and Wilkinson proved a classification theorem of smooth 1 BS(1, k)-action on RP 1 . As a corollary, they obtained the following rigidity result.…”

Section: Introductionmentioning

confidence: 99%

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