Polynomial growth harmonic functions on finitely generated abelian groups

2011/12/31 by Bobo Hua, Jürgen Jost, Juergen Jost +1
Mathematics · #Abelian group #Cayley graph #Dimension (graph theory) #Elementary abelian group #Generating set of a group #Geometric and Algebraic Topology #Geometry and complex manifolds #Harmonic analysis #Harmonic function #Harmonic measure #Holomorphic and Operator Theory #Polynomial #math.MG #msc:05C63 #msc:31C05 #msc:82B41

paper · pdf · doi:10.1007/s10455-013-9374-0

15 pages, to appear in Ann. Global Anal. Geom

openalex publication_date 2013/04/05 · arxiv created 2013/05/01 · arxiv updated 2013/05/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

In the present paper, we develop geometric analytic techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We develop a geometric analytic proof of the classical Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic functions on lattices \mathdsZⁿ that does not use a representation formula for harmonic functions. We also calculate the precise dimension of the space of polynomial growth harmonic functions on finitely generated abelian groups. While the Cayley graph not only depends on the abelian group, but also on the choice of a generating set, we find that this dimension depends only on the group itself.

Citations

• A Course in Metric Geometry
• Harmonic functions on complete riemannian manifolds
• Differential equations on riemannian manifolds and their geometric applications
• Ricci curvature and eigenvalue estimate on locally finite graphs
• A Course in the Theory of Groups