A Rigidity Property of Some Negatively Curved Solvable Lie Groups

TL;DR

This paper demonstrates that certain negatively curved solvable Lie groups have the property that all self-quasiisometries are almost isometries, by analyzing boundary maps and visual metrics.

Contribution

It establishes that all self-quasisymmetric maps of the boundary are bilipschitz, linking boundary behavior to the group's quasiisometric rigidity.

Findings

Self-quasiisometries are almost isometries for some negatively curved solvable Lie groups.

All self-quasisymmetric boundary maps are bilipschitz with respect to the visual metric.

Introduces parabolic visual metrics and relates them to visual metrics in hyperbolic spaces.

Abstract

We show that for some negatively curved solvable Lie groups, all self quasiisometries are almost isometries. We prove this by showing that all self quasisymmetric maps of the ideal boundary (of the solvable Lie groups) are bilipschitz with respect to the visual metric. We also define parabolic visual metrics on the ideal boundary of Gromov hyperbolic spaces and relate them to visual metrics.