Non‐realizability of the Torelli group as area‐preserving homeomorphisms

Chen, Lei; Marković, Vladimir

doi:10.1112/jlms.12340

Cited by 3 publications

(8 citation statements)

References 19 publications

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“…Since the rotation number of E(T c ) on the prime ends of K (E) is an irrational number r , it is semiconjugate to an irrational rotation. Then, up to the same semiconjugacy, the image of the centralizer of T c under E is SO (2). The image of each element is determined by its rotation number.…”

Section: 2mentioning

confidence: 99%

“…We know from Theorem 2.10 that the left and right prime end rotation numbers of K (E) are both r . In the group of circle homeomorphisms, the centralizer of an irrational rotation is essentially SO (2).…”

Section: The Proof Of Theorem 12mentioning

confidence: 99%

“…Comparing with the method in [2], the novelty of this paper is to provide a different proof towards the final contradiction of the result in [2]. The original contradiction is to use the fact that a certain Dehn twist is a product of commutators in its centralizer.…”

Section: Introductionmentioning

confidence: 99%

“…In the beginning of §4, we present an outline of the proof. Since this paper has a lot of overlap with [2], we omit or sketch many proofs to reduce redundancy.…”

Section: Introductionmentioning

confidence: 99%

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Non-realizability of the pure braid group as area-preserving homeomorphisms

Chen

2020

Ergod. Th. Dynam. Sys.

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Let $\operatorname{Homeo}_{+}(D_{n}^{2})$ be the group of orientation-preserving homeomorphisms of $D^{2}$ fixing the boundary pointwise and $n$ marked points as a set. The Nielsen realization problem for the braid group asks whether the natural projection $p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo}_{+}(D_{n}^{2}))$ has a section over subgroups of $B_{n}$ . All of the previous methods use either torsion or Thurston stability, which do not apply to the pure braid group $PB_{n}$ , the subgroup of $B_{n}$ that fixes $n$ marked points pointwise. In this paper, we show that the pure braid group has no realization inside the area-preserving homeomorphisms using rotation numbers.

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Section: 2mentioning

confidence: 99%

Section: The Proof Of Theorem 12mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In the beginning of §4, we present an outline of the proof. Since this paper has a lot of overlap with [2], we omit or sketch many proofs to reduce redundancy.…”

Section: Introductionmentioning

confidence: 99%

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Non-realizability of the pure braid group as area-preserving homeomorphisms

Chen

2020

Ergod. Th. Dynam. Sys.

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“…Franks-Handel [FH09], Bestvina-Church-Suoto [BCS13] and Salter-Tshishiku [ST16] have also obtained the nonrealization theorems for C 1 -diffeomorphisms. Recently, Chen [Che19] proved that braid group has no realization in the group of homeomorphisms and Chen-Markovic [CM20] proved that the Torelli group has no realization in the group of area-preserving homeomorphisms. The current big open problem in this area is whether every finite index subgroup of the mapping class group has realization as homeomorphisms or not.…”

Section: Introductionmentioning

confidence: 99%

Non-realizability of some big mapping class groups

Chen¹,

He²

2021

Preprint

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Non-realizability of the pure braid group as area-preserving homeomorphisms

Chen¹

2020

Preprint

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Let Homeo+(D 2 n ) be the group of orientation-preserving homeomorphisms of D 2 fixing the boundary pointwise and n marked points as a set. Nielsen realization problem for the braid group asks whether the natural projection pn : Homeo+(D 2 n ) → Bn := π0(Homeo+(D 2 n )) has a section over subgroups of Bn. All of the previous methods either use torsions or Thurston stability, which do not apply to the pure braid group P Bn, the subgroup of Bn that fixes n marked points pointwise. In this paper, we show that the pure braid group has no realization inside the areapreserving homeomorphisms using rotation numbers.

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Non‐realizability of the Torelli group as area‐preserving homeomorphisms

Cited by 3 publications

References 19 publications

Non-realizability of the pure braid group as area-preserving homeomorphisms

Non-realizability of the pure braid group as area-preserving homeomorphisms

Non-realizability of some big mapping class groups

Non-realizability of the pure braid group as area-preserving homeomorphisms

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