Coarse differentiation and quasi-isometries of a class of solvable Lie groups I

Abstract

This is the first of two papers that aim to understand quasi-isometries of a class of unimodular split solvable Lie groups. In the present paper, we show that locally (in a coarse sense), a quasi-isometry between two groups in this class is close to a map that respects their group structures. In the sequel to this paper [11], we will use this result to show quasi-isometric rigidity.