  In this paper we study certain groups of bilipschitz maps of the boundary
minus a point of a negatively curved space that is an abelian-by-cyclic
solvable Lie group, where the extension is given by a matrix whose eigenvalues
all lie outside of the unit circle. The case where the extension matrix is
diagonal was previously studied by Dymarz. 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.
