A Tukia-type theorem for nilpotent Lie groups and quasi-isometric rigidity of solvable groups

Abstract

In this paper we study uniform quasiconformal groups of Carnot-by-Carnot groups. We show that they can be conjugated into conformal groups provided the induced action on the space of distinct pairs is cocompact. Following the approach of Eskin-Fisher-Whyte [17] these results have applications to quasi-isometric rigidity of certain families of solvable groups.