Birkbeck, University of London, Wednesday November 5th, 2014
Room 745, Malet Street
If you wish to attend, please contact b.fairbairn [at] bbk.ac.uk
A rational number is a strong approximation for an irrational number if there is no closer rational number with smaller denominator. It is well known that the strong approximants of an irrational number are given by the convergents of its simple continued fraction expansion. In this talk we look at how the idea of strong approximation can be generalised to the approximation of irrational numbers by fixed points of parabolic elements of Fuchsian groups, taking a geometric approach. In particular, we will show that strong approximants by parabolic fixed points of the Theta group are precisely the convergents of even-integer continued fractions.
Recently S. Chmutov introduced an operation on maps on surfaces called partial duality which generalises the usual notion of map duality. Partial duals preserve connectedness and the number of edges, but not necessarily the number of vertices and faces, and therefore the genus of the embedding is not necessarily preserved. Using the combinatorial model for maps on surfaces we will look at how partial duality affects the monodromy and automorphism groups of the original map. For example, we can trivially observe that a map will have abelian monodromy if and only if its partial duals have abelian monodromy.
The recent landmark results of Kahn and Markovic have completed an epic voyage (steered largely by the amazing insights of Bill Thurston) through geometric topology in dimensions 2 and 3, and we now know in principle everything about the structure of compact hyperbolic 3-manifolds: they are virtual surface bundles with pseudo-Anosov monodromy. This talk will survey some of the background and try to answer the obvious question: what (necessarily finite) automorphism groups can occur for these 3-manifolds?