Abelian and solvable subgroups of the mapping class groups

Birman, Joan S.; Lubotzky, Alexander; McCarthy, John

doi:10.1215/s0012-7094-83-05046-9

Cited by 217 publications

(250 citation statements)

References 19 publications

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“…In this section we prove Theorem A.2. Let M be a multicurve in S g consisting entirely of nonseparating curves, and let f be an arbitrary multitwist supported in M ; without loss of generality, we assume that M is the canonical reduction system [7] for f ; that is, we assume that each curve of M is "used" by f . To prove the theorem, we need to show that f gives rise to a nontrivial element of Out(Γ/Γ 3 ); that is, if ϕ is any representative of f that preserves the basepoint, then we can find an element ξ of π 1 (S g ) so that ϕ (ξ)ξ −1 is not an element of Γ 3 .…”

mentioning

confidence: 99%

The dimension of the Torelli group

Bestvina

Bux

Margalit

2009

J. Amer. Math. Soc.

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We prove that the cohomological dimension of the Torelli group for a closed, connected, orientable surface of genus g ≥ 2 g \geq 2 is equal to 3 g − 5 3g-5 . This answers a question of Mess, who proved the lower bound and settled the case of g = 2 g=2 . We also find the cohomological dimension of the Johnson kernel (the subgroup of the Torelli group generated by Dehn twists about separating curves) to be 2 g − 3 2g-3 . For g ≥ 2 g \geq 2 , we prove that the top dimensional homology of the Torelli group is infinitely generated. Finally, we give a new proof of the theorem of Mess that gives a precise description of the Torelli group in genus 2. The main tool is a new contractible complex, called the “complex of minimizing cycles”, on which the Torelli group acts.

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confidence: 99%

The dimension of the Torelli group

Bestvina

Bux

Margalit

2009

J. Amer. Math. Soc.

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“…Kerckhoff [50] solved the Nielsen realization problem by showing convexity properties of length functions in Teichmüller space. A strong form of the Tits alternative was established in [15,59] and a classification of subgroups by Ivanov in [43].…”

Section: Thurston and Beyondmentioning

confidence: 99%

Book Review: A primer on mapping class groups

Bestvina¹

2014

Bull. Amer. Math. Soc.

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“…Birman, Lubotzky, and McCarthy [13] showed that every solvable subgroup of the mapping class group is virtually abelian, and that a free abelian subgroup of Mod(S g ) has rank at most 3g − 3. Moreover, they gave a very clear picture of what free abelian subgroups look like: each is (up to finite index) contained in the free abelian group generated by a collection of pseudo-Anosov maps and Dehn twists supported on disjoint subsurfaces.…”

Section: Constructions Of Pseudo-anosov Mapsmentioning

confidence: 99%

Book Review: Thurston’s work on surfaces

Margalit¹

2013

Bull. Amer. Math. Soc.

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In 1976 (incidentally, the year the reviewer was born), Thurston first circulated his now-famous preprint on the classification of surface homeomorphisms [81]. It states that every homeomorphism of a compact surface is homotopic to a homeomorphism in standard form.The generic standard form is a pseudo-Anosov homeomorphism. Such a homeomorphism acts on the surface by preserving two transverse singular measured foliations, multiplying the measure of one by a stretch factor λ > 1 and the other by 1/λ. In other words, away from a finite set of singular points, a pseudo-Anosov homeomorphism is modeled on an Anosov map of the torus, or equivalently, an element of SL(2, Z) with two real eigenvalues.The precise statement of Thurston's theorem is: Every homeomorphism of a compact surface is homotopic to a homeomorphism that either (a) has finite order, (b) is reducible (that is, fixes an essential 1-submanifold), or (c) is pseudoAnosov. Moreover, the last case is exclusive from the first two. It was realized after Thurston's work (see [63]) that Nielsen had assembled all of the tools needed for the classification [66][67][68][69], and so this theorem is sometimes called the Nielsen-Thurston classification theorem.Let Mod(S) be the mapping class group of a surface S, that is, the group of homotopy classes of homeomorphisms of S. Another way to state the classification which suggests an analogy with the Jordan canonical form is: for every element of Mod(S), there is a representative φ and a (possibly empty) φ-invariant 1-submanifold C so the restriction of φ to S − C has two kinds of components, finite order and pseudo-Anosov. Birman, Lubotzky, and McCarthy showed that the reduction system C is canonical [13].Shortly after Thurston's announcement, there was a flurry of activity to understand what he did. The book under review is the product of a year-long seminar at Orsay devoted to Thurston's work. The lectures were based on notes from Thurston's graduate course at Princeton University, handwritten by Michael Handel and William Floyd.In the broadest of strokes, Thurston's proof proceeds as follows. Let S g be a closed, connected, orientable surface of genus g ≥ 2. Fricke showed that the Teichmüller space Teich(S g ), the space of hyperbolic metrics on S g up to isotopy, is an open ball of dimension 6g − 6. The key idea of Thurston is that Teich(S g ) has a compactification that is homeomorphic to a closed ball; the boundary sphere is PMF(S g ), the space of projective classes of measured foliations on S g . Moreover, Mod(S g ) acts on this closed ball. Then, in a most spectacular application of the Brouwer fixed point theorem, Thurston concludes that each element of Mod(S g ) fixes some point of Teich(S g ) ∪ PMF(S g ); by analyzing the various cases for the fixed point, we obtain the classification theorem.

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Abelian and solvable subgroups of the mapping class groups

Cited by 217 publications

References 19 publications

The dimension of the Torelli group

The dimension of the Torelli group

Book Review: A primer on mapping class groups

Book Review: Thurston’s work on surfaces

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