Bilipschitz maps of boundaries of certain negatively curved homogeneous spaces

TL;DR

This paper investigates bilipschitz maps on boundaries of certain negatively curved solvable Lie groups, advancing understanding of their geometric structure and contributing to the proof of quasi-isometric rigidity for specific lattice classes.

Contribution

It extends previous work by analyzing non-diagonal extension matrices, completing the proof of quasi-isometric rigidity for a broader class of solvable Lie group lattices.

Findings

Characterization of bilipschitz maps on boundary spaces

Completion of quasi-isometric rigidity proof for certain lattices

Extension of prior diagonal matrix results to more general cases

Abstract

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.