Tara E. Brendle

Braids: A Survey

Publisher Summary Crossings are suggested as they are in a picture of a highway overpass on a map. The identity braid has a canonical representation in which two strands never cross. Multiplication of braids is by juxtaposition, concatenation, isotopy, and rescaling. This chapter discusses Artins braid group, B n and its role in knot theory. The chapter illustrates ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups. Artins braid group is naturally isomorphic to the mapping class group of an n-times punctured disc. The chapter also illustrates the topological concept of a braid and of a group of braids via the notion of a configuration space. It then outlines the new developments in the area of B n mapping, thus illustrating how to pass from diffeomorphisms to geometric braids and back again..

Commensurations of the Johnson kernel.

Let K be the subgroup of the extended mapping class group, Mod(S), generated by Dehn twists about separating curves. Assuming that S is a closed, orientable surface of genus at least 4, we confirm a conjecture of Farb that Comm(K) � Aut(K) � Mod(S). More generally, we show that any injection of a finite index subgroup of K into the Torelli group I of S is induced by a homeomorphism. In particular, this proves that K is co-Hopfian and is characteristic in I. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of I into I is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes.

Configuration spaces of rings and wickets

The main result in this paper is that the space of all smooth links in R3 isotopic to the trivial link of n components has the same homotopy type as its finite-dimensional subspace consisting of configurations of n unlinked Euclidean circles (the ‘rings’ in the title). There is also an analogous result for spaces of arcs in upper half-space, with circles replaced by semicircles (the ‘wickets’ in the title). A key part of the proofs is a procedure for greatly reducing the complexity of tangled configurations of rings and wickets. This leads to simple methods for computing presentations for the fundamental groups of these spaces of rings and wickets as well as various interesting subspaces. The wicket spaces are also shown to be aspherical.

Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = 1

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at

Addendum to: Commensurations of the Johnson kernel

t=-1

Factoring in the hyperelliptic Torelli group

t=-1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added.

Point pushing, homology, and the hyperelliptic involution

Formanek and Procesi have demonstrated that Aut(Fn) is not linear for n 3. Their technique is to construct nonlinear groups of a special form, which we call FP-groups, and then to embed a special type of automorphism group, which we call a poison group ,i n Aut( F n), from which they build an FP-group. We rst prove that poison groups cannot be embedded in certain mapping class groups. We then show that no FP-groups of any form can be embedded in mapping class groups. Thus the methods of Formanek and Procesi fail in the case of mapping class groups, providing strong evidence that mapping class groups may in fact be linear. AMS Classication 57M07,20F65; 57N05,20F34

The Birman–Craggs–Johnson homomorphism and abelian cycles in the Torelli group

Let K.S/ be the subgroup of the extended mapping class group, Mod.S/, generated by Dehn twists about separating curves. In our earlier paper, we showed that Comm.K.S//a Aut.K.S//a Mod.S/ when S is a closed, connected, orientable surface of genus g 4. By modifying our original proof, we show that the same result holds for g 3, thus confirming Farb’s conjecture in all cases (the statement is not true for g 2). 20F36 The purpose of this note is to extend the results of our paper “Commensurations of the Johnson kernel” to the lone remaining case. We briefly review the notation and basic ideas before explaining the improvement. We refer the reader to our earlier paper for further details [1]. Let SDSg denote a closed, connected, orientable surface of genus g , and let Mod.S/ denote the extended mapping class group (orientation reversing elements are allowed). The Torelli group I.S/ is the subgroup of Mod.S/ consisting of elements that act trivially on H1.S;Z/, and the Johnson kernel K.S/ is the subgroup of I.S/ generated by Dehn twists about separating curves. The abstract commensurator of a group A , denoted Comm.A/, is the group of isomorphisms of finite index subgroups ofA (under composition), with two such isomorphisms equivalent if they agree on a finite index subgroup of A . The product of W G! H with W G 0 ! H 0 is a map defined on 1 .H\G 0 /.