Two questions on mapping class groups

Funar, Louis

doi:10.1090/s0002-9939-2010-10555-5

Cited by 10 publications

(12 citation statements)

References 18 publications

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“…This result improves a result of L. Funar [4] which shows that homomorphisms from the genus g ≥ 1 mapping class group to GL(n, C) have finite image provided n ≤ √ g + 1.…”

Section: Introduction and Statement Of Resultssupporting

confidence: 86%

Triviality of some representations of 𝑀𝐶𝐺(𝑆_{𝑔}) in 𝐺𝐿(𝑛,ℂ),𝔻𝕚𝕗𝕗(𝕊²) and ℍ𝕠𝕞𝕖𝕠(𝕋²)

Franks

Handel

2013

Proc. Amer. Math. Soc.

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We show the triviality of representations of the mapping class group of a genus g surface in GL(n, C), Diff(S 2 ) and Homeo(T 2 ) when appropriate restrictions on the genus g and the size of n hold. For example, if S g is a surface of finite type and φ : MCG(S g ) → GL(n, C) is a homomorphism, then φ is trivial provided the genus g ≥ 3 and n < 2g. We also show that if S g is a closed surface with genus g ≥ 7, then every homomorphism φ : MCG(S g ) → Diff(S 2 ) is trivial and that if g ≥ 3, then every homomorphism φ : MCG(S g ) → Homeo(T 2 ) is trivial.

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“…This result improves a result of L. Funar [4] which shows that homomorphisms from the genus g ≥ 1 mapping class group to GL(n, C) have finite image provided n ≤ √ g + 1.…”

Section: Introduction and Statement Of Resultssupporting

confidence: 86%

Triviality of some representations of 𝑀𝐶𝐺(𝑆_{𝑔}) in 𝐺𝐿(𝑛,ℂ),𝔻𝕚𝕗𝕗(𝕊²) and ℍ𝕠𝕞𝕖𝕠(𝕋²)

Franks

Handel

2013

Proc. Amer. Math. Soc.

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“…On the other hand, it has been also shown by Bridson (see [6]) that M g,n has only finitely many irreducible linear representations over any algebraically closed field, up to dimension (g + 1). Later, Funar [15] showed that there is no linear representation with infinite image up to dimension about √ g + 1. However, there is an obvious linear representation of rank 2g which comes from the action of M g,n on the homology of Σ g,0 (the map θ g,n defined below).…”

mentioning

confidence: 99%

Finite index subgroups of mapping class groups

Berrick

Gebhardt

Paris

2013

Proceedings of the London Mathematical Society

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Let g ≥ 3 and n ≥ 0, and let M g,n be the mapping class group of a surface of genus g with n boundary components. We prove that M g,n contains a unique subgroup of index 2 g−1 (2 g − 1) up to conjugation, a unique subgroup of index 2 g−1 (2 g + 1) up to conjugation, and the other proper subgroups of M g,n are of index greater than 2 g−1 (2 g +1). In particular, the minimum index for a proper subgroup of M g,n is 2 g−1 (2 g − 1). AMS Subject Classification. Primary: 57M99. Secondary: 20G40, 20E28. Introduction and statement of resultsThe interaction between mapping class groups and finite groups has long been a topic of interest. The famous Hurwitz bound of 1893 showed that a closed Riemann surface of genus g has an upper bound of 84(g − 1) for the order of its finite subgroups, and Kerckhoff showed that the order of finite cyclic subgroups is bounded above by 4g + 2 [18], [19]. The subject of finite index subgroups of mapping class groups was brought into focus by Grossman's discovery that the mapping class group M g,n = π 0 (Homeo(Σ g,n )) of an oriented surface Σ g,n of genus g and n boundary components is residually finite, and thus well-endowed with subgroups of finite index [16]. (Homeo(Σ g,n ) denotes the space of those homeomorphisms of Σ g,n that preserve the orientation and are the identity on the boundary.) This prompts the "dual" question: -What is the minimum index mi(M g,n ) of a proper subgroup of finite index in M g,n ? Results to date have suggested that, like the maximum finite order question, the minimum index question should have an answer that is linear in g. The best previously published bound is mi(M g,n ) > 4g + 4 for g ≥ 3 (see [25]). This inequality is used by Aramayona and Souto to prove that, if g ≥ 6 and g ′ ≤ 2g − 1, then any nontrivial homomorphism M g,n → M g ′ ,n ′ is induced by an embedding [1]. It is also an important ingredient in the proof of Zimmermann [33] that, for g = 3 and 4, the minimal nontrivial quotient of M g,0 is Sp 2g (F 2 ).The "headline" result of this paper is the following exact, exponential bound.Theorem 0.1. For g ≥ 3 and n ≥ 0, mi(M g,n ) = mi(Sp 2g (Z)) = mi(Sp 2g (F 2 )) = 2 g−1 (2 g − 1) .This exponential bound is all the more surprising since in similar questions we get linear (expected) bounds. For instance, Bridson [6,7] has proved that a mapping class group of a surface Finite index subgroups of mapping class groups September 26, 2018

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“…The first result on the nonexistence of faithful linear representations of the mapping class group was obtained by Farb-Lubotzky-Minsky [2] who proved that no homomorphism from a subgroup of finite index of the mapping class group into GL(n, C) is injective for n < 2 √ g − 1. This was improved first by Funar [8] who showed that every map from the mapping class group into SL(n, C) has finite image for n ≤ √ g + 1. This was improved further by…”

Section: Introduction and The Resultsmentioning

confidence: 97%