Automorphisms of Teichmüller modular groups

Ivanov, N. V.

doi:10.1007/bfb0082778

Cited by 49 publications

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“…In this section we discuss the following result, due to Ivanov [46] and McCarthy [71], and a few of its extensions: Theorem 3.4 (Ivanov, McCarthy). Let X be a surface of genus at least 3.…”

Section: Automorphisms and Injective Endomorphismsmentioning

confidence: 99%

“…This analogy has motivated many, possibly most, advances in the understanding of the mapping class group. For example, Grossman [35] proved that Map(X) is residually finite; Birman, Lubotzky and McCarthy [16] proved that the Tits alternative holds for subgroups of Map(X); the Thurston classification of elements in Map(X) mimics the classification of elements in an algebraic group [91]; Harvey [37] introduced the curve complex in analogy with the Tits' building; Harer's [36] computation of the virtual cohomological dimension of Map(X) follows the outline of Borel and Serre's argument for arithmetic groups [18], etc... On the other hand, the comparison between Map(X) and SL n Z has strong limitations; for instance the mapping class group contains many infinite normal subgroups of infinite index [25], has finite index in its abstract commensurator [46], and has infinite dimensional second bounded cohomology [15]. In addition, it is not known if the mapping class group contains finite index subgroups Γ with H 1 (Γ; R) = 0.…”

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confidence: 99%

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Rigidity phenomena in the mapping class group

Aramayona

Souto

2016

Handbook of Teichmüller Theory, Volume VI

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Throughout this article we will consider connected orientable surfaces of negative Euler characteristic and of finite topological type, meaning of finite genus and with finitely many boundary components and/or cusps. We will feel free to think about cusps as marked points, punctures or topological ends. Sometimes we will need to make explicit mention of the genus and number of punctures of a surface: in this case, we will write S g,n for the surface of genus g with n punctures and empty boundary. Finally, we define the complexity of a surface X as the number κ(X) = 3g − 3 + p, where g is the genus and p is the number of cusps and boundary components of X.In order to avoid too cumbersome notation, we denote byf is an orientation-preserving homeomorphism fixing pointwise the boundary and each puncture of X    the group of orientation-preserving self-homeomorphisms of X relative to the boundary and the set of punctures. We endow Homeo(X) with the compactopen topology, and denote by Homeo 0 (X) the connected component of the identity Id : X → X. It is well-known that Homeo 0 (X) consists of those elements in Homeo(X) that are isotopic to Id : X → X relative to ∂X and the set of punctures of X. The mapping class group Map(X) of X is the group Map(X) = Homeo(X)/ Homeo 0 (X). In the literature, Map(X) is sometimes referred to as the pure mapping class group. We will also need to consider the extended mapping class group Map * (X), i.e. the group of all isotopy classes of self-homeomorphisms of X. Note that if X has r boundary components and n punctures, we have an exact sequencewhere Sym s is the group of permutations of the set with s elements. Let T (X) and M(X) = T (X)/ Map(X) be, respectively, the Teichmüller and moduli spaces of X. The triad formed by Map(X), T (X) and M(X) is often compared with the one formed, for n ≥ 3, by SL n Z, the symmetric space SO n \ SL n R, and the locally symmetric space SO n \ SL n R/ SL n Z.The second author has been partially supported by NSERC Discovery and Accelerator Supplement grants.

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“…In this section we discuss the following result, due to Ivanov [46] and McCarthy [71], and a few of its extensions: Theorem 3.4 (Ivanov, McCarthy). Let X be a surface of genus at least 3.…”

Section: Automorphisms and Injective Endomorphismsmentioning

confidence: 99%

mentioning

confidence: 99%

Rigidity phenomena in the mapping class group

Aramayona

Souto

2016

Handbook of Teichmüller Theory, Volume VI

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“…It follows from work of Birman-Lubotzky-McCarthy that for any surface F , rk Mod(F ) is realized by any subgroup generated by powers of Dehn twists about curves forming a pants decomposition for F [7]; thus, rk Mod(S m ) = m − 3. The following theorem of Ivanov gives another connection between the algebra and topology of Mod(S m ) [24]. We restrict our attention here to the genus 0 case, which has a particularly simple statement.…”

Section: and Powers Of Dehn Twists Commute If And Only If The Curvementioning

confidence: 99%

Injections of Artin groups

Bell

Margalit

2007

Comment. Math. Helv.

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Abstract. We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a sphere with at least 5 punctures. In particular, we show that any injective homomorphism between these groups is given by a homeomorphism of a punctured sphere together with a map to the integers. The technique, following Ivanov, is to prove that every superinjective map of the curve complex of a sphere with at least 5 punctures is induced by a homeomorphism. We also determine the automorphism group of the pure braid group on at least 4 strands. Mathematics Subject Classification (2000). Primary 20F36; Secondary 57M07.

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“…Theorem 6 was inspired by the theorem of Ivanov [Iv1], [Iv2] (see also [McC]) to the effect that Out(Mod ± (S)) = 1 for genus(S) ≥ 3.…”

Section: Theorem 3 (T G(s) Is Connected) For a Closed Orientable Surmentioning

confidence: 99%

The Torelli geometry and its applications

Farb

Ivanov

2005

Mathematical Research Letters

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Let S be a closed orientable surface of genus g. The mapping class group Mod(S) of S is defined as the group of isotopy classes of orientationpreserving diffeomorphisms S → S. We will need also the extended mapping class group Mod ± (S) of S which is defined as the group of isotopy classes of all diffeomorphisms S → S. Let us fix an orientation of S. Then the algebraic intersection number provides a nondegenerate, skew-symmetric, bilinear form on H = H 1 (S, Z), called the intersection form. The natural action of Mod(S) on H preserves the intersection form. If we fix a symplectic basis in H, then we can identify the group of symplectic automorphisms of H with the integral symplectic group Sp(2g, Z) and the action of Mod(S) on H leads to a natural surjective homomorphismThe Torelli group of S, denoted by I S , is defined as the kernel of this homomorphism; that is, I S is the subgroup of Mod(S) consisting of elements which act trivially on H 1 (S, Z). In particular, we have the following wellknown exact sequenceThe group I S plays an important role both in algebraic geometry and in low-dimensional topology. At the same time most of the basic questions *

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Automorphisms of Teichmüller modular groups

Cited by 49 publications

References 6 publications

Rigidity phenomena in the mapping class group

Rigidity phenomena in the mapping class group

Injections of Artin groups

The Torelli geometry and its applications

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