Coarse differentiation of quasi-isometries II: Rigidity for Sol and Lamplighter groups

In this paper, which is the continuation of [EFW2], we complete the proof of the quasi-isometric rigidity of Sol and the lamplighter groups. The results were announced in [EFW1].

💡 Research Summary

This paper completes the quasi‑isometric rigidity program for the solvable Lie group Sol and for lamplighter groups, building on the authors’ earlier work (EFW1, EFW2). The central tool is the technique of coarse differentiation, a method that extracts linear‑like behavior from large‑scale quasi‑isometries. By applying this technique to the two model spaces, the authors show that any quasi‑isometry of Sol or of a lamplighter group is at bounded distance from an actual group automorphism, thereby proving rigidity.

For Sol, the space is modeled as ℝ³ equipped with the left‑invariant metric ds² = e^{2z}dx² + e^{-2z}dy² + dz². The authors introduce a height function (the z‑coordinate) and a family of horizontal planes. Coarse differentiation yields, on sufficiently large balls, an approximation of any quasi‑isometry by a map of the form (x, y, z) ↦ (A·x + B·y, C·x + D·y, z + τ) where the 2×2 matrix