Isomorphisms Between Big Mapping Class Groups

Bavard, Juliette; Dowdall, Spencer; Rafi, Kasra

doi:10.1093/imrn/rny093

Cited by 28 publications

(53 citation statements)

References 18 publications

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“…Informally, these results imply that mapping class groups and Out(F N ) do not have natural enveloping 'Lie groups'. These strong rigidity results have recently been extended to other groups, such as handlebody groups [Hen18] and big mapping class groups [BDR18].…”

Section: Introductionmentioning

confidence: 95%

Commensurations of subgroups of 𝑂𝑢𝑡(𝐹_{𝑁})

Horbez

Wade

2020

Trans. Amer. Math. Soc.

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A theorem of Farb and Handel [FH07] asserts that for N ≥ 4, the natural inclusion from Out(F N ) into its abstract commensurator is an isomorphism. We give a new proof of their result, which enables us to generalize it to the case where N = 3. More generally, we give sufficient conditions on a subgroup Γ of Out(F N ) ensuring that its abstract commensurator Comm(Γ) is isomorphic to its relative commensurator in Out(F N ). In particular, we prove that the abstract commensurator of the Torelli subgroup IA N for all N ≥ 3, or more generally any term of the Andreadakis-Johnson filtration if N ≥ 4, is equal to Out(F N ). Likewise, if Γ the kernel of the natural map from Out(F N ) to the outer automorphism group of a free Burnside group of rank N ≥ 3, then the natural map Out(F N ) → Comm(Γ) is an isomorphism.

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Section: Introductionmentioning

confidence: 95%

Commensurations of subgroups of 𝑂𝑢𝑡(𝐹_{𝑁})

Horbez

Wade

2020

Trans. Amer. Math. Soc.

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“…There are also more general theorems characterizing isomorphisms-and injective maps-between different complexes; see for instance the work of Aramayona [2], Aramayona-Leininger [3], Bavard-Dowdall-Rafi [4], Birman-Broaddus-Menasco [5], Hernández [27,28], Irmak [29,31], and Shackleton [57]. Pathologies.…”

Section: 2mentioning

confidence: 99%

Normal subgroups of mapping class groups and the metaconjecture of Ivanov

Brendle

Margalit

2019

J. Amer. Math. Soc.

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We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group. These results resolve the metaconjecture of N.V. Ivanov, which asserts that any "sufficiently rich" object associated to a surface has automorphism group isomorphic to the extended mapping class group, for a broad class of such objects. As applications, we show: (1) right-angled Artin groups and surface groups cannot be isomorphic to normal subgroups of mapping class groups containing elements of small support, (2) normal subgroups of distinct mapping class groups cannot be isomorphic if they both have elements of small support, and (3) distinct normal subgroups of the mapping class group with elements of small support are not isomorphic. Our results also suggest a new framework for the classification of normal subgroups of the mapping class group.2000 Mathematics Subject Classification. Primary: 20F36; Secondary: 57M07.

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“…Subsequently, Bavard-Dowdall-Rafi [2] established the analogous result for every infinite-type surface. In a similar fashion, Farb-Ivanov [9], Brendle-Margalit [6,7,5], and Kida [16] proved that, for all but a few finite-type surfaces, Comm I(Σ) ∼ = Aut I(Σ) ∼ = MCG(Σ).…”

Section: Torelli Complexmentioning

confidence: 80%

“…The following statement follows from [2]. The first assertion is Lemma 2.6, while the second follows from its proof.…”

Section: 2mentioning

confidence: 86%

“…In order to prove the theorem, we closely follow Brendle-Margalit's strategy. First, we adapt ideas of Bavard-Dowdall-Rafi [2] to show that every commensuration of the Torelli group respects the property of being a separating twist or a bounding pair map. From this we deduce that every commensuration induces an automorphism of a combinatorial object called the Torelli complex.…”

Section: Introductionmentioning

confidence: 99%

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Big Torelli groups: generation and commensuration

et al. 2019

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For any surface Σ of infinite topological type, we study the Torelli subgroup I(Σ) of the mapping class group MCG(Σ), whose elements are those mapping classes that act trivially on the homology of Σ. Our first result asserts that I(Σ) is topologically generated by the subgroup of MCG(Σ) consisting of those elements in the Torelli group which have compact support. In particular, using results of Birman [4], Powell [24], and Putman [25] we deduce that I(Σ) is topologically generated by separating twists and bounding pair maps. Next, we prove the abstract commensurator group of I(Σ) coincides with MCG(Σ). This extends the results for finite-type surfaces [9,6,7,16] to the setting of infinite-type surfaces.

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Isomorphisms Between Big Mapping Class Groups

Cited by 28 publications

References 18 publications

Commensurations of subgroups of 𝑂𝑢𝑡(𝐹_{𝑁})

Commensurations of subgroups of 𝑂𝑢𝑡(𝐹_{𝑁})

Normal subgroups of mapping class groups and the metaconjecture of Ivanov

Big Torelli groups: generation and commensuration

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