Mapping Class Groups

Ivanov, N. V.

doi:10.1016/b978-044482432-5/50013-5

Cited by 125 publications

(157 citation statements)

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“…Not only are the Γ g Poincaré duality groups, but they are also fundamental groups of closed aspherical manifolds, with base Γ g−1 and fiber UT (S g−1 ). The proof of this stronger statement is given by Ivanov [26,Section 6.3].…”

Section: Theorem 92 (Relative Birman Exact Sequence) Let G ≥mentioning

confidence: 99%

“…The work of Ivanov [26,Corollary 1.8] shows that the map from Stab I(S g ) (M ) to Mod(S g − M ) has image in PMod(S g − M ), and so for each i there is a map Stab I(S g ) (M ) − → PMod(R i ); in particular, the R i are preserved. LetR i be the surface obtained from R i by forgetting all of the punctures, and letR i be the surface obtained from R i by forgetting all of the punctures but one.…”

Section: Theorem 610 (Vautaw) For M As Above We Have G(m ) ∩ I(smentioning

confidence: 99%

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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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Section: Theorem 92 (Relative Birman Exact Sequence) Let G ≥mentioning

confidence: 99%

Section: Theorem 610 (Vautaw) For M As Above We Have G(m ) ∩ I(smentioning

confidence: 99%

The dimension of the Torelli group

Bestvina

Bux

Margalit

2009

J. Amer. Math. Soc.

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“…These fixed points behave like attracting and repelling fixed points for φ. More specifically, with s ∈ {+1, −1}, for any neighbourhood U of λ s in T (Σ) and any compact set K in T (Σ) \ {λ −s } we have φ sn (K) ⊆ U for sufficiently large n (see [11]). It is known that a pseudo-Anosov mapping class φ fixes a bi-infinite Teichmüller geodesic, the axis of φ, on which it acts by translation.…”

Section: Mapping Class Groupsmentioning

confidence: 99%

“…We refer the reader to [2], [9] and [11] for detailed studies of geodesic and measured geodesic laminations, Teichmüller spaces and mapping class groups, respectively, and recall only what we need. Throughout this paper, a surface Σ will mean an orientable connected surface of negative Euler characteristic, with genus g and p punctures, and with empty boundary.…”

Section: Introductionmentioning

confidence: 99%

Free subgroups of surface mapping class groups

Anderson

Aramayona

Shackleton

2007

Conform. Geom. Dyn.

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“…The mapping class group MCG(Σ) associated to Σ is the group of all homotopy classes of orientation preserving self-homeomorphisms of Σ. (For a thorough account of these groups, we refer the reader to Ivanov [13].) It is known that every mapping class group MCG(Σ) is finitely presentable and can be generated by Dehn twists.…”

Section: Mapping Class Groupsmentioning

confidence: 99%