Large scale geometry of certain solvable Lie groups

Tullia Dymarz, Yale University

Abstract: In their proof of quasi-isometric rigidity for lattices in SOL, Eskin-Fisher-Whyte develop a technique of "coarse differentiation" to prove a theorem on the structure of quasi-isometries of SOL. This structure theorem combined with a theorem on bilipshitz maps of the real line completes the proof of quasi-isometric rigidity for lattices in SOL. Eskin-Fisher-Whyte and Peng use similar techniques to extend this structure theorem to quasi-isometries of more general solvable Lie groups. We will present the ingredients needed to complete the proof of quasi-isometric rigidity for these more general solvable Lie groups. This involves, among other things, a rigidity theorem on the boundaries of certain negatively curved homogeneous spaces.