Rigidity of mapping class group actions on
S1

Mann, Kathryn; Wolff, Maxime

doi:10.2140/gt.2020.24.1211

Cited by 21 publications

(14 citation statements)

References 17 publications

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“…This group acts (by point-pushing) on the ideal boundary of the fundamental group of the surface; one calls this the geometric action. Mann and Wolff [13] showed that every non-trivial action of g,1 on the circle is semiconjugate to the geometric action when g > 2. Their proof and ours are quite dissimilar, but there is some conceptual overlap.…”

Section: Introductionmentioning

confidence: 99%

Big mapping class groups and rigidity of the simple circle

Calegari¹,

Chen²

2020

Ergod. Th. Dynam. Sys.

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Section: Introductionmentioning

confidence: 99%

Big mapping class groups and rigidity of the simple circle

Calegari¹,

Chen²

2020

Ergod. Th. Dynam. Sys.

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“…Even though we are only able to prove that these actions cannot be faithful, the conclusion of Theorem 1.1 and the results in [11] suggest that "the kernel of the action should be large".…”

Section: Introductionmentioning

confidence: 80%

“…In a recent article, the authors in [11] were able to show that any C 1 action of Mod(Σ g,1 ), where Σ g,1 is a surface of genus at least 3 and has exactly one marked point (puncture), must be trivial. So, actions of the full mapping class group on the circle are fairly well understood, and the results in this paper are the first step towards understanding C 1 actions of finite index subgroups of mapping class groups.…”

Section: Introductionmentioning

confidence: 99%

$C^1$ actions on the circle of finite index subgroups of $Mod(Σ_g)$, $Aut(F_n)$, and $Out(F_n)$

Parwani¹

2021

Preprint

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Let Σ g be a closed, connected, and oriented surface of genus g ≥ 24 and let Γ be a finite index subgroup of the mapping class group M od(Σ g ) that contains the Torelli group I(Σ g ). Then any orientation preserving C 1 action of Γ on the circle cannot be faithful.We also show that if Γ is a finite index subgroup of Aut(F n ), when n ≥ 8, that contains the subgroup of IA-automorphisms, then any orientation preserving C 1 action of Γ on the circle cannot be faithful.Similarly, if Γ is a finite index subgroup of Out(F n ), when n ≥ 8, that contains the Torelli group T n , then any orientation preserving C 1 action of Γ on the circle cannot be faithful.In fact, when n ≥ 10, any orientation preserving C 1 action of a finite index subgroup of Aut(F n ) or Out(F n ) on the circle cannot be faithful.

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“…As a consequence, the notion of weak conjugacy arises naturally in settings where one needs to consider continuous conjugacy invariants. This is for example the case in the study of mapping class group actions on the circle; see the article by K. Mann and M. Wolff [41] and the references therein.…”

Section: Rokhlin Property and Hamiltonian Homeomorphismmentioning

confidence: 99%

Barcodes and area-preserving homeomorphisms

Roux,

Seyfaddini,

Viterbo

2018

Preprint

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In this paper we use the theory of barcodes as a new tool for studying dynamics of area-preserving homeomorphisms. We will show that the barcode of a Hamiltonian diffeomorphism of a surface depends continuously on the diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms.Our main dynamical application concerns the notion of weak conjugacy, an equivalence relation which arises naturally in connection to C 0 continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for a large class of Hamiltonian homeomorphisms with a finite number of fixed points, the number of fixed points, counted with multiplicity, is a weak conjugacy invariant. The proof relies, in addition to the theory of barcodes, on techniques from surface dynamics such as Le Calvez's theory of transverse foliations.In our exposition of barcodes and persistence modules, we present a proof of the Isometry Theorem which incorporates Barannikov's theory of simple Morse complexes.

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Rigidity of mapping class group actions on S1

Cited by 21 publications

References 17 publications

Big mapping class groups and rigidity of the simple circle

Big mapping class groups and rigidity of the simple circle

$C^1$ actions on the circle of finite index subgroups of $Mod(Σ_g)$, $Aut(F_n)$, and $Out(F_n)$

Barcodes and area-preserving homeomorphisms

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