36 research outputs found
Normal subgroups of mapping class groups and the metaconjecture of Ivanov
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.Comment: 57 pages, 11 figure
Configuration spaces of rings and wickets
The main result in this paper is that the space of all smooth links in
Euclidean 3-space 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 K(G,1)'s.Comment: 28 pages. Some revisions in the expositio
The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary
We give two proofs that appropriately defined congruence subgroups of the
mapping class group of a surface with punctures/boundary have enormous amounts
of rational cohomology in their virtual cohomological dimension. In particular
we give bounds that are super-exponential in each of three variables: number of
punctures, number of boundary components, and genus, generalizing work of
Fullarton-Putman. Along the way, we give a simplified account of a theorem of
Harer explaining how to relate the homotopy type of the curve complex of a
multiply-punctured surface to the curve complex of a once-punctured surface
through a process that can be viewed as an analogue of a Birman exact sequence
for curve complexes.
As an application, we prove upper and lower bounds on the coherent
cohomological dimension of the moduli space of curves with marked points. For
, we compute this coherent cohomological dimension for any number of
marked points. In contrast to our bounds on cohomology, when the surface has marked points, these bounds turn out to be independent of , and
depend only on the genus.Comment: 29 pages, 3 figures; some small correction
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
Online assessment and feedback: how to square the circle
Within Mathematics considerable changes have taken place in how we teach, assess and provide feedback to the very large cohorts of students that make up our non-honours classes. The use of online assessment methods have been introduced to provide enhanced feedback β both in terms of frequency and volume β but this raises huge logistical problems: how to process and record over 2000 individually marked pieces of work each week.
This presentation will describe how an innovative use of online assessment methods, scanning technology and conventional marking can all be integrated with the University's SharePoint system, resulting in a system where rapid, weekly feedback can be both given and recorded. These changes
have resulted in increased student engagement and improvement in exam performance.
Many of the ideas and processes that have been developed will be transferable to other schools. For example, the dynamic use of SharePoint to record weekly assessed work has enabled the School to develop effective early warning mechanisms to identify students with problems