Radial isoperimetry and absence of harmonic functions with ℓ^p-gradient

2025/09/25 by Gournay, Antoine
#05C81 #20F6 #20J06 #31C05 #43A07 #60G50 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2509.22731

Abstract

In this paper we show that groups for which the probability of return of a random walk is bounded below by K₁ exp(-K₂n^c) have no non-constant harmonic functions with gradient in ℓ^p. The proof relies on results from ℓ^p-cohomology, a form of radial isoperimetry, transport patterns and revisiting some results of Følner.

Citations

• Cuts, flows and gradient conditions on harmonic functions
• Exact descriptions of Følner functions and sets on wreath products and Baumslag-Solitar groups
• Separation profiles, isoperimetry, growth and compression
• A finitary version of Gromov's polynomial growth theorem

Related

• Absence of non-constant harmonic functions with ℓ^p-gradient in some semi-direct products
• Cuts, flows and gradient conditions on harmonic functions
• Boundaries and harmonic functions for random walks with random transition probabilities
• The ratio set of the harmonic measure of a random walk on a hyperbolic group
• Speed of random walks, isoperimetry and compression of finitely generated groups