7 research outputs found
The virtual Haken conjecture: Experiments and examples
A 3-manifold is Haken if it contains a topologically essential surface. The
Virtual Haken Conjecture says that every irreducible 3-manifold with infinite
fundamental group has a finite cover which is Haken. Here, we discuss two
interrelated topics concerning this conjecture.
First, we describe computer experiments which give strong evidence that the
Virtual Haken Conjecture is true for hyperbolic 3-manifolds. We took the
complete Hodgson-Weeks census of 10,986 small-volume closed hyperbolic
3-manifolds, and for each of them found finite covers which are Haken. There
are interesting and unexplained patterns in the data which may lead to a better
understanding of this problem.
Second, we discuss a method for transferring the virtual Haken property under
Dehn filling. In particular, we show that if a 3-manifold with torus boundary
has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of
the Dehn filled manifolds are virtually Haken. We use this to show that every
non-trivial Dehn surgery on the figure-8 knot is virtually Haken.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper12.abs.htm
Model selection in the space of Gaussian models invariant by symmetry
We consider multivariate centred Gaussian models for the random variable
, invariant under the action of a subgroup of the group of
permutations on . Using the representation theory of the
symmetric group on the field of reals, we derive the distribution of the
maximum likelihood estimate of the covariance parameter and also the
analytic expression of the normalizing constant of the Diaconis-Ylvisaker
conjugate prior for the precision parameter . We can thus
perform Bayesian model selection in the class of complete Gaussian models
invariant by the action of a subgroup of the symmetric group, which we could
also call complete RCOP models. We illustrate our results with a toy example of
dimension and several examples for selection within cyclic groups,
including a high dimensional example with .Comment: 34 pages, 4 figures, 5 table
Fischer-Clifford theory for split and non-split group extensions.
Thesis (Ph.D)-University of Natal, Pietermaritzburg, 2001.The character table of a finite group provides considerable amount of information about the group, and hence is of great importance in Mathematics as well as in Physical Sciences. Most of the maximal subgroups of the finite simple groups and their automorphisms are of extensions of elementary abelian groups, so methods have been developed for calculating the character tables of extensions of elementary abelian groups. Character tables of finite groups can be constructed using various techniques. However Bernd Fischer presented a powerful and interesting technique for calculating the character tables of group extensions. This technique, which is known as the technique of the Fischer-Clifford matrices, derives its fundamentals from the Clifford theory. If G=N.G is an appropriate extension of N by G, the method involves the construction of a nonsingular matrix for each conjugacy class of G/N~G. The character table of G can then be determined from these Fischer-Clifford matrices and the character table of certain subgroups of G, called inertia factor groups. In this dissertation, we described the Fischer-Clifford theory and apply it to both split and non-split group extensions. First we apply the technique to the split extensions 2,7:Sp6(2) and 2,8:SP6(2) which are maximal subgroups of Sp8(2) and 2,8:08+(2) respectively. This technique has also been discussed and used by many other researchers, but applied only to split extensions or to the case when every irreducible character of N can be extended to an irreducible character of its inertia group in G. However the same method can not be used to construct character tables of certain non-split group extensions. In particular, it can not be applied to the non-split extensions of the forms 3,7.07(3) and 3,7.(0,7(3):2) which are maximal subgroups of Fischer's largest sporadic simple group Fi~24 and its automorphism group Fi24 respectively. In an attempt to generalize these methods to such type of non-split group extensions, we need to consider the projective representations and characters. We have shown that how the technique of Fischer-Clifford matrices can be applied to any such type of non-split extensions. However in order to apply this technique, the projective characters of the inertia factors must be known and these can be difficult to determine for some groups. We successfully applied the technique of Fischer-Clifford matrices and determined the Fischer-Clifford matrices and hence the character tables of the non-split extensions 3,7.0,7(3) and 3,7.(0,7(3):2). The character tables computed in this thesis have been accepted for incorporation into GAP and will be available in the latest versions