Characterisations of algebraic properties of groups in terms of harmonic functions

Tointon, Matthew

doi:10.4171/ggd/375

Cited by 10 publications

(5 citation statements)

References 22 publications

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“…Remark 1.7. Some of the material we use to prove Theorem 1.6 originally appeared in an early version of the third author's paper [21], in which the bound dim H k (G, µ) ≥ dim P k (N )− dim P k−1 (N ) was obtained (the notation here differs slightly from the notation there).…”

Section: 2mentioning

confidence: 99%

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Polynomials and harmonic functions on discrete groups

Meyerovitch

Perl

Tointon

et al. 2016

Trans. Amer. Math. Soc.

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Alexopoulos proved that on a finitely generated virtually nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free nilpotent subgroup of finite index is always a polynomial in the Mal'cev coordinates of that subgroup. For general groups, vanishing of higher-order discrete derivatives gives a natural notion of polynomial maps, which has been considered by Leibman and others. We provide a simple proof of Alexopoulos's result using this notion of polynomials, under the weaker hypothesis that the space of harmonic functions of polynomial growth of degree at most k is finite dimensional. We also prove that for a finitely generated group the Laplacian maps the polynomials of degree k surjectively onto the polynomials of degree k − 2. We then present some corollaries. In particular, we calculate precisely the dimension of the space of harmonic functions of polynomial growth of degree at most k on a virtually nilpotent group, extending an old result of Heilbronn for the abelian case, and refining a more recent result of Hua and Jost.2010 Mathematics Subject Classification. 20F65 (primary), 05C25 (secondary).

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Section: 2mentioning

confidence: 99%

“…The second is to compute precisely the dimensions of the subspaces of harmonic functions of polynomial growth of given degree. This computation is in turn an ingredient in a recent proof of the third author that the dimension of the space of all harmonic functions is finite if and only if G is virtually cyclic [21].…”

Section: Introductionmentioning

confidence: 99%

Polynomials and harmonic functions on discrete groups

Meyerovitch

Perl

Tointon

et al. 2016

Trans. Amer. Math. Soc.

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“…Subsequently to writing a preliminarily version of our results, the construction above was exploited and generalized in a different direction by Tointon [28] to characterize groups with the property that the space of all harmonic functions is finite dimensional. Also, after finishing this paper we observed that a somewhat related construction of positive harmonic functions on affine groups appears in [2,6].…”

Section: Introductionmentioning

confidence: 99%

Harmonic functions of linear growth on solvable groups

Meyerovitch

Yadin

2016

Isr. J. Math.

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Abstract. In this work we study the structure of finitely generated groups for which a space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse direction to Kleiner's theorem). We prove that this is indeed the case for solvable groups. The investigation is partly motivated by Kleiner's proof for Gromov's theorem on groups of polynomial growth.

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“…Such harmonic functions do not appear to contribute to the discussion of the Liouville property (see, for example, [24, Defn 2.1.10]), since both their positive and negative parts are unbounded. For recent articles on the related aspect of geometric group theory, the reader is referred to [30,35]. This paper is organized as follows.…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

Connective constants and height functions for Cayley graphs

Grimmett

2017

Trans. Amer. Math. Soc.

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Abstract. The connective constant µ(G) of an infinite transitive graph G is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support so-called 'unimodular graph height functions'. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular graph height function termed here a 'group height function'. A necessary and sufficient condition for the existence of a group height function is presented, and may be applied in the context of the bridge constant, and of the locality of connective constants for Cayley graphs. Locality may thereby be established for a variety of infinite groups including those with strictly positive deficiency.It is proved that a large class of Cayley graphs support unimodular graph height functions, that are in addition harmonic on the graph. This implies, for example, the existence of unimodular graph height functions for the Cayley graphs of finitely generated solvable groups. It turns out that graphs with non-unimodular automorphism subgroups also possess graph height functions, but the resulting graph height functions need not be harmonic.Group height functions, as well as the graph height functions of the previous paragraph, are non-constant harmonic functions with linear growth and an additional property of having periodic differences. The existence of such functions on Cayley graphs is a topic of interest beyond their applications in the theory of self-avoiding walks.

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Characterisations of algebraic properties of groups in terms of harmonic functions

Cited by 10 publications

References 22 publications

Polynomials and harmonic functions on discrete groups

Polynomials and harmonic functions on discrete groups

Harmonic functions of linear growth on solvable groups

Connective constants and height functions for Cayley graphs

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