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.
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.
The dimension of the Torelli group BESTVINA, MLADEN; BUX, KAI-UWE; MARGALIT, DAN
Journal of the American Mathematical Society,
01/2010, Letnik:
23, Številka:
1
Journal Article
Recenzirano
Odprti dostop
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.
In the 1970s, Joan Birman and Hugh Hilden wrote several papers on the problem of relating the mapping class group of a surface to that of a covering space. Their results provide a bridge between the ...theories of mapping class groups and braid groups. We survey the work of Birman and Hilden, give an overview of the subsequent developments, and discuss open questions and new directions.
We show that any two elements of the pure braid group either commute or generate a free group, settling a question of Luis Paris. Our proof involves the theory of 3-manifolds and the theory of group ...actions on trees.
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