Normal subgroups of mapping class groups and the metaconjecture of Ivanov

Brendle, Tara E.; Margalit, Dan

doi:10.1090/jams/927

Cited by 27 publications

(48 citation statements)

References 47 publications

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“…Theorem 1.1 applies to mapping classes whose supports are 'sufficiently large'. On the other hand, a result of Brendle and the third author roughly states that the normal closure of any mapping class with 'sufficiently small' support is not isomorphic to a RAAG [BM19,Corollary 1.4]. This leads us to the following conjecture.…”

Section: Which Raags?mentioning

confidence: 99%

“…In the absence of freeness, we may hope that the normal closure of a power of is isomorphic to a RAAG, that is, a group where all of the defining relations are commutations among generators. However, Brendle and the third author [BM19] showed that if the support of a mapping class is sufficiently small (in a precise sense that they define) then its normal closure is not isomorphic, or even abstractly commensurable, to a RAAG; see also [CLM14].…”

Section: Introductionmentioning

confidence: 99%

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Right-angled Artin groups as normal subgroups of mapping class groups

Clay¹,

Mangahas²,

Margalit³

2021

Compositio Math.

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We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.

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Section: Which Raags?mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Right-angled Artin groups as normal subgroups of mapping class groups

Clay¹,

Mangahas²,

Margalit³

2021

Compositio Math.

Self Cite

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“…This includes the Torelli group [FI14] (with a recent extension to big mapping class groups in [AGK + 18]), or more generally the further terms of the Johnson filtration [BM04, Kid13,BPSon]. The latest development is a result by Brendle and Margalit [BM17], asserting that if Γ is a normal subgroup of Mod(Σ g ) that contains a 'small' element (roughly, a homeomorphism supported on at most one third of the surface), then the natural map Mod ± (Σ g ) → Comm(Γ) induced by conjugation is an isomorphism. We warn the reader that the condition on 'small' elements cannot be removed, as Mod(Σ g ) also contains normal purely pseudo-Anosov free subgroups [DGO17], and as recalled earlier the abstract commensurator of a nonabelian free group is not finitely generated.…”

Section: Introductionmentioning

confidence: 99%

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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“…In our paper [24] we conjecture that there is a stronger statement than the one suggested by item (1) in Problem 7.4. Specifically, we conjecture that the small assumption can be removed altogether, if we keep the assumption of connectivity.…”

Section: Ivanov's Metaconjecturementioning

confidence: 66%

Problems, questions, and conjectures about mapping class groups

Margalit¹

2019

Proceedings of Symposia in Pure Mathematics

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We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudo-Anosov stretch factors, Torelli subgroups, and normal subgroups.Beginning with the work of Max Dehn a century ago, the subject of mapping class groups has become a central topic in mathematics. It enjoys deep and varied connections to many other subjects, such as lowdimensional topology, geometric group theory, dynamics, Teichmüller theory, algebraic geometry, and number theory. The number of papers on mapping class groups recorded on MathSciNet in the last six decades has grown from 205 to 386 to 525 to 791 to 1,121 to 1,390. The subject seems to enjoy an endless supply of beautiful ideas, pictures, and theorems. arXiv:1806.08773v1 [math.GT] 22 Jun 2018 Question 1.1. Is Mod(S g ) linear?Dehn proved that Mod(T 2 ), the mapping class group of the torus, is isomorphic to SL 2 (Z) ⊆ GL 2 (C). In the case g = 2 the Birman-Hilden theory [108] gives a short exact sequence:where ι is the hyperelliptic involution of S 2 . The group Mod(S 0,6 ) is closely related to the braid group on 5 strands. As such, Bigelow-Budney and Korkmaz were able to prove that Mod(S 2 ) is linear, using the theorem of Krammer and Bigelow that braid groups are linear [14,90]. For g ≥ 3, Question 1.1 is wide open.Linearity of the braid group. One might be tempted to think that Mod(S g ) is not linear, because if it were then we would already have stumbled across the representation. On the other hand, we should draw inspiration from the case of the braid group. The Burau representation

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Normal subgroups of mapping class groups and the metaconjecture of Ivanov

Cited by 27 publications

References 47 publications

Right-angled Artin groups as normal subgroups of mapping class groups

Right-angled Artin groups as normal subgroups of mapping class groups

Commensurations of subgroups of 𝑂𝑢𝑡(𝐹_{𝑁})

Problems, questions, and conjectures about mapping class groups

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