Growth rates of groups, harmonic functions and the strong Liouville property

2024/05/13 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.2405.07688

Abstract

We study the interplay between the growth of positive harmonic functions, the strong Liouville property and the large scale geometry of finitely generated groups. We first show that the existence of one non-constant minimal harmonic function of slow (respectively, fast) growth rate guarantees that the space spanned by minimal harmonic functions of slow (respectively, fast) growth is infinite dimensional. We give a geometric-analytic characterisation for the strong Liouville property in terms of a functional involving the Green's function, which has immediate applications to groups with reasonable heat kernel bounds, including groups of polynomial as well as exponential growth. On the way, we show that under certain assumptions, strong Liouville property implies that the Green speed is zero.

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