Quantitative versions of Pansu Asymptotic Theorem and of Mitchell Tangent Theorem

TL;DR

This paper provides quantitative estimates on how quickly geodesic Lie groups converge to their metric limits, improving bounds for nilpotent and general cases and enhancing understanding of their asymptotic behavior.

Contribution

It introduces new quantitative bounds on the convergence rates of geodesic Lie groups to their tangent or asymptotic metrics, refining previous results.

Findings

Sharper bounds on metric convergence rates

Quantitative estimates for nilpotent geodesic Lie groups

Improved understanding of tangent metric approximation

Abstract

We quantitatively study the speed of convergence of geodesic Lie groups to their metric limits. For nilpotent geodesic Lie groups, we give estimates on the difference of the original metrics and the asymptotic metrics, while for general geodesic Lie groups, we give similar estimates for the difference of the original metrics and the tangent metrics. In both settings, our results sharpen existing bounds in the literature.