23 research outputs found
Factoring in the hyperelliptic Torelli group
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. The authors and
Putman 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
elements.Comment: 9 pages, 7 figure
Braids: A Survey
This article is about Artin's braid group and its role in knot theory. We set
ourselves two goals: (i) to provide enough of the essential background so that
our review would be accessible to graduate students, and (ii) to focus on those
parts of the subject in which major progress was made, or interesting new
proofs of known results were discovered, during the past 20 years. A central
theme that we try to develop is to show ways in which structure first
discovered in the braid groups generalizes to structure in Garside groups,
Artin groups and surface mapping class groups. However, the literature is
extensive, and for reasons of space our coverage necessarily omits many very
interesting developments. Open problems are noted and so-labelled, as we
encounter them.Comment: Final version, revised to take account of the comments of readers. A
review article, to appear in the Handbook of Knot Theory, edited by W.
Menasco and M. Thistlethwaite. 91 pages, 24 figure
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) and Mod(S) are all isomorphic. 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.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper37.abs.htm
The level four braid group
By evaluating the Burau representation at t=-1, we obtain a symplectic
representation of the braid group. We study the resulting congruence subgroups
of the braid group, namely, the preimages of the principal congruence subgroups
of the symplectic group. Our main result is that the level 4 congruence
subgroup is equal to the group generated by squares of Dehn twists. We also
show that the image of the Brunnian subgroup of the braid group under the
symplectic representation is the level four congruence subgroup.Comment: 17 pages, 4 figures; minor corrections to the published versio
Every mapping class group is generated by 6 involutions
Let Mod_{g,b} denote the mapping class group of a surface of genus g with b
punctures. Feng Luo asked in a recent preprint if there is a universal upper
bound, independent of genus, for the number of torsion elements needed to
generate Mod_{g,b}. We answer Luo's question by proving that 3 torsion elements
suffice to generate Mod_{g,0}. We also prove the more delicate result that
there is an upper bound, independent of genus, not only for the number of
torsion elements needed to generate Mod_{g,b} but also for the order of those
elements. In particular, our main result is that 6 involutions (i.e.
orientation-preserving diffeomorphisms of order two) suffice to generate
Mod_{g,b} for every genus g >= 3, b = 0, and g >= 4, b = 1.Comment: 15 pages, 7 figures; slightly improved main result; minor revisions.
to appear in J. Al
Homomorphisms of Two Bridge Knot Groups onto Special Linear Groups
The enumeration of knots is a problem central to knot theory; It focuses on the question of whether two given knots are in some sense the same. Two different means of defining a knot and equivalence of knots will be fully described and then shown to be equivalent. We then create an algebraic invariant which we can apply to the problem\ud
of knot enumeration. Finally, a topological understanding of this\ud
algebraic tool will be developed