2025/12/07 by Mukherjee, Mayukh, Samanta, Soumyadeb, Thandar, Soumyadip
#FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #Probability (math.PR)
paper · doi:10.48550/arxiv.2512.06753
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+φ([x]). For any finite generating set S this yields a canonical isometric identification LHF(G,μ)/ℂ ≅ Hom(Gab,ℂ), ‖∇S f‖_∞=maxs∈ S|φ([s])|, independent of the choice of centered measure. Next, for any finite-index subgroup H≤ G and adapted smooth μ we prove a quantitative induction-restriction principle: restriction along H and an explicit averaging operator give a linear isomorphism LHF(G,μ)\congLHF(H,μH), where μH 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 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.