Quasiconformal homogeneity of hyperbolic surfaces

Recall that a complete hyperbolic manifold is uniformly
quasiconformally homogeneous if there exists a $K \geq 1$ so that every
pair of points on the manifold can be paired by a $K$-quasiconformal
automorphism of the manifold. We will review the definition, some
basic results and some geometric and topological constraints associated
with quasiconformal homogeneity. Then we will focus on quasiconformal
homogeneity in dimension $2$, which has to be dealt with quite
differently from the higher-dimensional case. We will present recent
results for hyperelliptic surfaces and planar domains.