ABSTRACT: A set $A\subset\hat\mathbb{C}$ is $K$-quasiconformally homogeneous if for any two points $x,y\in A$ there exists a $K$-quasiconformal mapping $f:\hat\mathbb{C}\to\hat\mathbb{C}$ that keeps $A$ setwise invariant and maps $x$ to $y$. In particular, we will focus on planar domains such that (a) the domain itself is quasiconformally homogeneous, (b) the domain’s boundary is quasiconformally homogemeous, or (c) both the domain and its boundary are quasiconformally homogeneous. We explore relations between these classes of domains as well as their geometric characterizations.

This is joint work with Edward Taylor.