The Lipschitz Liouville Property, Affine Rigidity, and Coarse Harmonic Coordinates on Groups of Polynomial Growth

TL;DR

This paper develops a comprehensive theory of Lipschitz harmonic functions on groups of polynomial growth, establishing affine rigidity, invariance under quasi-isometries, and a discrete-to-continuous extension framework.

Contribution

It introduces new rigidity results, invariance principles, and extension theorems for Lipschitz harmonic functions on groups of polynomial growth, advancing understanding of their geometric and analytic properties.

Findings

Lipschitz harmonic functions on nilpotent groups are affine functions.

LHF is a quasi-isometry invariant for groups with polynomial growth.

Discrete Lipschitz harmonic data can be extended to continuous harmonic functions with controlled gradients.

Abstract

We develop a quantitative theory of Lipschitz harmonic functions (LHF) on finitely generated groups, with emphasis on the Lipschitz Liouville property, affine rigidity, and quasi-isometric invariance for groups of polynomial growth. On finitely generated nilpotent groups we prove an affine rigidity theorem: for any adapted, smooth, Abelian-centered probability measure $\mu$, every Lipschitz $\mu$-harmonic function is affine, $f(x)=c+\varphi([x])$. For any finite generating set $S$ this yields a canonical isometric identification $$ \mathrm{LHF}(G,\mu)/\mathbb{C} \cong \mathrm{Hom}(G_{\mathrm{ab}},\mathbb{C}),\qquad \|\nabla_S f\|_\infty=\max_{s\in S}|\varphi([s])|, $$ independent of the choice of centered measure. Next, for any finite-index subgroup $H\le G$ and adapted smooth $\mu$ we prove a quantitative induction-restriction principle: restriction along $H$ and an explicit averaging operator give a linear isomorphism $\mathrm{LHF}(G,\mu)\cong\mathrm{LHF}(H,\mu_H)$, where $\mu_H$ is the hitting measure, with two-sided control of the Lipschitz seminorms. For groups of polynomial growth equipped with SAS measures we then show that $\mathrm{LHF}$ is a quasi-isometry invariant, and use this to construct coarse harmonic coordinates that straighten quasi-isometries up to bounded error. Finally, within the Lyons-Sullivan / Ballmann-Polymerakis discretization framework, we prove a quantitative discrete-to-continuous extension theorem: Lipschitz harmonic data on an orbit extend to globally Lipschitz $L$-harmonic functions on the ambient manifold, with gradient bounds controlled by the background geometry.