The Hilbert-Smith conjecture for quasiconformal actions.
Quasiregular semigroups.
Triangle subgroups of Kleinian groups.
Infinite group actions on spheres.
This paper is mainly intended as a survey of the recent work of a number of authors concerning certain infinite group actions on spheres and to raise some as yet unanswered questions. The main thrust of the current research in this area has been to decide what topological and geometrical properties characterise the infinite conformal or Möbius groups. One should then obtain reasonable topological or geometrical restrictions on a subgroup G of the homeomorphism group of a sphere which will imply...
Riesz transforms and related singular integrals.
Squeezing the Sierpinski sponge
We give an example relating to the regularity properties of mappings with finite distortion. This example suggests conditions to be imposed on the distortion function in order to avoid "cavitation in measure".
Quasiconformal groups of compact type.
We establish that a quasiconformal group is of compact type if and only if its limits set is purely conical and find that the limit set of a quasiconformal group of compact type is uniformly perfect. A key tool is the result of Bowditch-Tukia on compact-type convergence groups. These results provide crucial tools for studying the deformations of quasiconformal groups and in establishing isomorphisms between such groups and conformal groups.
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