On harmonic functions and the linear-growth case of Gromov's theorem

2013/01/08 by Tointon, Matthew
#Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1301.1566

Abstract

We show that the space of harmonic functions on a finitely generated infinite group G is finite dimensional if, and only if, G has a finite-index subgroup isomorphic to the integers. A key tool is Wilkie and van den Dries's quantitative version of the linear-growth case of Gromov's theorem on groups of polynomial growth.

Related

• Harmonic functions of linear growth on solvable groups
• Polynomially growing harmonic functions on connected groups
• A new proof of Gromov's theorem on groups of polynomial growth
• A functional analysis proof of Gromov's polynomial growth theorem
• Harmonic functions on locally compact groups of polynomial growth