Abstract Liouville’s theorem says that in dimension greater than two, all conformal maps are Möbius transformations. We prove an analogous statement about simplicial complexes, where two complexes considered discretely conformally equivalent if they combinatorially and the lengths of corresponding edges related by scale factors associated with vertices.

In this paper, an analogue of Bieberbach’s distortion theorem has been proposed, in a class quasi-conformal maps.