Groups with minimal harmonic functions as small as you like

Gideon Amir; Gady Kozma; Nicolás Matte Bon

doi:10.4171/GGD/748

Groups with minimal harmonic functions as small as you like

Gideon Amir, Gady Kozma, Nicolás Matte Bon

Bar-Ilan University, Weizmann Institute of Science

Research output: Contribution to journal › Article › peer-review

Abstract

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 Z/2 ≀_X Γ in which the base group Γ is defined via its properly chosen action on X .

Original language	English
Pages (from-to)	1-24
Number of pages	24
Journal	Groups, Geometry, and Dynamics
Volume	18
Issue number	1
DOIs	https://doi.org/10.4171/GGD/748
State	Published - 13 Feb 2024

Keywords

Harmonic functions
Schreier graphs
group actions
random walks

All Science Journal Classification (ASJC) codes

Geometry and Topology
Discrete Mathematics and Combinatorics

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10.4171/GGD/748

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abstract = "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 Z/2 ≀X Γ in which the base group Γ is defined via its properly chosen action on X .",

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AB - 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 Z/2 ≀X Γ in which the base group Γ is defined via its properly chosen action on X .

KW - Harmonic functions

KW - Schreier graphs

KW - group actions

KW - random walks

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Groups with minimal harmonic functions as small as you like

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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