Groups with minimal harmonic functions as small as you like (with an appendix by Nicolás Matte Bon). (English) Zbl 1540.20085
Summary: For any order of growth \(f(n)=o(\log n)\), we construct a finitely-generated group \(G\) and a set of generators \(S\) such that the Cayley graph of \(G\) with respect to \(S\) supports a harmonic function with growth \(f\) but does not support any harmonic function with slower growth. The construction uses permutational wreath products \(\mathbb{Z}/2 \wr_X \Gamma\) in which the base group \(\Gamma\) is defined via its properly chosen action on \(X\).
MSC:
| 20F69 | Asymptotic properties of groups |
| 20P05 | Probabilistic methods in group theory |
| 60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |
| 20E22 | Extensions, wreath products, and other compositions of groups |
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