The sub-Riemannian affine-additive group
Elia Bubani
The affine-additive group $\mathcal{AA}$ consists of $\mathbb{R}\times\mathcal{R}$, where $\mathbb{R}$ is the real line and $\mathcal{R}$ is the hyperbolic right half-plane, endowed with a natural group law. We consider $\mathcal{AA}$ as a metric measure space with a canonical left-invariant measure and a left-invariant sub-Riemannian metric. Throughout the talk we shall cover notions of parabolicity and hyperbolicity for metric measure spaces and then point out implications in the theory of quasiconformal mappings. As a consequence we obtain that $\mathcal{AA}$ is not quasiconformally equivalent to the sub-Riemannian Heisenberg group $\mathbb{H}$. If time allows we will discuss further relations between $\mathcal{AA}$ and $\mathbb{H}$ under the more general theory of quasiregular mappings.
Joint work with Z. Balogh and G. Platis.