Reconstruction of domains from their groups of quasiconformal autohomeomorphisms

Abstract

For an open set U ⊆ R n , let QC( U) denote the group of all quasiconformal homeo morphism of U. The following is our first main result. Let U ⊆ R m and V ⊆ R n be open, and suppose that τ is a group isomorphism between QC( U) and QC( V). Then there is a quasiconformal homeomorphism ϑ from U onto V such that ϕ induce τ. That is, for every - f ϵ QC( U): τ(-) = ϑ - to - to ϑ −1.