Bilipschitz maps of boundaries of certain negatively curved homogeneous spaces

Abstract

In this paper we study certain groups of bilipschitz maps of the boundary minus a point of a negatively curved space of the form R x M R-n, where M is a matrix whose eigenvalues all lie outside of the unit circle. The case where M is diagonal was previously studied by Dymarz (Geom Funct Anal (GAFA) 19:1650-1687, 2009). As an application, combined with work of Eskin-Fisher-Whyte and Peng, we provide the last steps in the proof of quasi-isometric rigidity for a class of lattices in solvable Lie groups.