This document last changed 25 July 2003.

In July 2003, Al Marden, Vlad Markovic and I submitted Complex earthquakes and deformations of the unit disk This paper contains a new, conceptual, proof of Sullivan's theorem that there is a uniform quasiconformal constant for quasiconformal homeomorphisms between an open simply-connected proper subset of the plane and the boundary of the hyperbolic convex hull of its complement. It also contains an approach to McMullen's Disk Theorem that works without the presence of a group. This approach is by no means fully worked out, and a lot remains to be done. Finally, we tackle the problem of finding explicitly the substantial boundary dilatation at any point of the boundary of a euclidean convex domain, for quasiconformal maps which are homotopic to the nearest point retraction (the homotopy keeping the boundary fixed). We show that the nearest point retraction for a euclidean convex region in the plane is at most two.

I have been working for some time with Al Marden and Vlad Markovic on the convex hull of a compact subset of the 2-sphere in hyperbolic 3-space, and the work referred to above follows on from the work I describe here. David Wright gave some important help with drawing relevant pictures. Click for a description by David Wright of his pictures.

Some of our results have been accepted by the Annals of Mathematics. During the next few days I hope to add the most recent versions of these papers to this website, and also to update this website which is now full of obsolete remarks and links.

In the meantime, slightly older versions are referred to below. You can probably download recent versions from Vlad Markovic's website http://www.maths.warwick.ac.uk/~markovic.

A paper accepted by Annals, is unfortunately rather huge from the point of view of downloading. This is mainly because the counter-example to Thurston's equivariant K=2 conjecture is a rather large picture. Click here for a version with no pictures.

Our work started with the theory of angle doubling, though the submitted paper referred to above does not use this theory. A somewhat obsolete version of this theory appears in our doubling paper. A new and improved version of this paper is in preparation, and will be inserted here in due course. The current, obsolete version of this paper contains many of the results which appear in submitted paper, but sometimes with proofs that are less accurate and are also substantially longer.

Recently we used Mathematica to show that the logarithmic spiral gives
a counter-example to the Thurston K=2 conjecture. Here we compute the
minimal qc constant K varying over all qc homeomorphisms, not just
those that commute with Moebius transformations preserving the region.
We find a region where the constant is greater than 2.1. You can
download our Mathematica program
in Notebook format, or a postscript
version, if you don't have Mathematica.

Knuth-Bendix for groups with infinitely many rules with Paul Sanders. Now published in IJAC in a slightly abbreviated form. The full version is on the xxx archives. The paper introduces a new class of groups for which the word problem is solvable reasonably quickly. There is a close connection with automatic groups, and in fact automatic groups give a special case of these groups.