Hilbert-Smith Conjecture for K - Quasiconformal Groups

MSC 2010: 30C60

Modulus method and radial stretch map in the Heisenberg group

We develop a modulus method for surface families inside a domain in the Heisenberg group and we prove that the stretch map between two Heisenberg spherical rings is a minimiser for the mean distortion among the class of contact quasiconformal maps between these rings which satisfy certain boundary conditions.

Uniqueness of minimisers for a Grötzsch-Belinskiĭ type inequality in the Heisenberg group

The modulus method introduced by H. Grötzsch yields bounds for a mean distortion functional of quasiconformal maps between two annuli mapping the respective boundary components onto each other. P. P. Belinskiĭ studied these inequalities in the plane and identified the family of all minimisers. Beyond the Euclidean framework, a Grötzsch-Belinskiĭ-type inequality has been previously considered for quasiconformal maps between annuli in the Heisenberg group whose boundaries are Korányi spheres. In this note we show that--in contrast to the planar situation--the minimiser in this setting is essentially unique.

Socialising across channels : group multichannel communication

People increasingly communicate over multiple channels, such as SMS, email and IM. Choosing the channel for interaction is typically a considered action and shapes the message itself. In order to explore how people make sense of communication mediums and more generally, social group behaviour, we developed a multichannel communication prototype. Preliminary results indicate that multichannel communication was considered very useful in the group context even considering the increased quantity of messages while it was little used for person-to-person interaction.