The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many ...important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students.
Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. This ...text brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics.
We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support, then its automorphism group and abstract commensurator group ...are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group. These results resolve the metaconjecture of N. V. Ivanov, which asserts that any ``sufficiently rich'' object associated to a surface has automorphism group isomorphic to the extended mapping class group, for a broad class of such objects. As applications, we show: (1) right-angled Artin groups and surface groups cannot be isomorphic to normal subgroups of mapping class groups containing elements of small support, (2) normal subgroups of distinct mapping class groups cannot be isomorphic if they both have elements of small support, and (3) distinct normal subgroups of the mapping class group with elements of small support are not isomorphic. Our results also suggest a new framework for the classification of normal subgroups of the mapping class group.
This book provides a detailed exposition of William Thurston's work on surface homeomorphisms, available here for the first time in English. Based on material of Thurston presented at a seminar in ...Orsay from 1976 to 1977, it covers topics such as the space of measured foliations on a surface, the Thurston compactification of Teichmüller space, the Nielsen-Thurston classification of surface homeomorphisms, and dynamical properties of pseudo- Anosov diffeomorphisms. Thurston never published the complete proofs, so this text is the only resource for many aspects of the theory.Thurston was awarded the prestigious Fields Medal in 1982 as well as many other prizes and honors, and is widely regarded to be one of the major mathematical figures of our time. Today, his important and influential work on surface homeomorphisms is enjoying continued interest in areas ranging from the Poincaré conjecture to topological dynamics and low-dimensional topology.Conveying the extraordinary richness of Thurston's mathematical insight, this elegant and faithful translation from the original French will be an invaluable resource for the next generation of researchers and students.
We give a complete classification of homomorphisms from the commutator subgroup of the braid group on n$n$ strands to the braid group on n$n$ strands when n$n$ is at least 7. In particular, we show ...that each non‐trivial homomorphism extends to an automorphism of the braid group on n$n$ strands. This answers four questions of Vladimir Lin. Our main new tool is the theory of totally symmetric sets.
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The proof
of the first statement of Theorem 5.1 of the paper referenced in the title is correct for
and incorrect for
and should be considered an open problem. As such, the proof of the second ...statement is not correct for
The dimension of the Torelli group BESTVINA, MLADEN; BUX, KAI-UWE; MARGALIT, DAN
Journal of the American Mathematical Society,
01/2010, Volume:
23, Issue:
1
Journal Article
Peer reviewed
Open access
We prove that the cohomological dimension of the Torelli group for a closed, connected, orientable surface of genus g \geq 2 is equal to 3g-5. This answers a question of Mess, who proved the lower ...bound and settled the case of 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 2g-3. For 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.
Factoring in the hyperelliptic Torelli group BRENDLE, TARA E.; MARGALIT, DAN
Mathematical proceedings of the Cambridge Philosophical Society,
09/2015, Volume:
159, Issue:
2
Journal Article
Peer reviewed
Open access
The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic ...involution. Putman and the authors proved that this group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. In this paper, we introduce an algorithmic approach to factoring a wide class of elements of the hyperelliptic Torelli group into such Dehn twists, and apply our methods to several basic types of elements. As one consequence, we answer an old question of Dennis Johnson.
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