Factoring Quasiconformal & quasisymmetric mappings

It follows from the Measurable Riemann Mapping Theorem that we can always present a $2$-dimensional quasi-conformal mapping as a composition of quasi-conformal mappings with smaller dilatation. In this talk we will construct $n (\ge 3)$-dimensional quasi-conformal homeomorphism between Euclidean spaces which admit no minimal factorization in linear, inner, or outer dilatation. If time permits, I will discuss the composition of quasi-symmetric mappings between metric spaces.