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

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 $μ$, every Lipschitz $μ$-harmonic function is affine, $f(x)=c+\varphi([x])$. For any finite generating set $S$ this yields a canonical isometric identification $$ \mathrm{LHF}(G,μ)/\mathbb{C} \cong \mathrm{Hom}(G_{\mathrm{ab}},\mathbb{C}),\qquad \|\nabla_S f\|_\infty=\max_{s\in S}|\varphi([s])|, $$ independent of the choice of centered measure. Next, for any finite-index subgroup $H\le G$ and adapted smooth $μ$ we prove a quantitative induction-restriction principle: restriction along $H$ and an explicit averaging operator give a linear isomorphism $\mathrm{LHF}(G,μ)\cong\mathrm{LHF}(H,μ_H)$, where $μ_H$ 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.