Polynomials and harmonic functions on discrete groups

Meyerovitch, Tom; Perl, Idan; Tointon, Matthew; Yadin, Ariel

doi:10.48550/arxiv.1505.01175

Cited by 3 publications

(5 citation statements)

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“…One consequence of our results is a structure theorem for the space of linearly growing harmonic functions for general groups where this space is finite dimensional: Up to an additive constant and passing to a finite index subgroups any such function must be a homomophism into the additive group R. In a follow-up paper joint with Idan Perl and Matthew Tointon [22] we provide, along with additional results, a structure theorem for the space of harmonic functions of polynomial growth (in the finite-dimensional case).…”

Section: Introductionmentioning

confidence: 86%

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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Section: Introductionmentioning

confidence: 86%

Harmonic functions of linear growth on solvable groups

Meyerovitch

Yadin

2016

Isr. J. Math.

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“…. , f d } spans H 1 (G, µ) is then precisely the linear-growth case of [2, Theorem 1.12]; see also [20] for a more elementary proof. shows that the bound 2M |T | on the dimension of the space of harmonic functions in the proof of Proposition 7.1 can be tight.…”

Section: Harmonic Functions On Virtually Abelian Groupsmentioning

confidence: 99%

“…The function we construct in proving Proposition 10.2 is positive, and so Proposition 8.3 implies that G has an infinite-dimensional space spanned by positive harmonic functions, although we do not need this to prove Theorem 1.1. It also implies that H 1 (G, µ) is infinite dimensional, since if dim H 1 (G, µ) < ∞ then every linearly growing harmonic function restricts to a homomorphism on some finite-index subgroup of G [20].…”

Section: Harmonic Functions On Groups With Virtually Cyclic Quotientsmentioning

confidence: 99%

“…Suppose first that G has a finite-index infinite nilpotent subgroup N of rank d ∈ N (the rank is defined in [20], for example, and is equal to 1 if and only if N is virtually cyclic). It follows from [20] that dim H k (G, µ) ≫ d k d−1 , which implies one direction of the theorem, the other direction being Proposition 7.1. Now suppose that G is not virtually nilpotent.…”

Section: Groups With Finite-dimensional Spaces Of Harmonic Functionsmentioning

confidence: 99%

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Characterisations of algebraic properties of groups in terms of harmonic functions

Tointon¹

2016

Groups Geom. Dyn.

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We prove various results connecting structural or algebraic properties of graphs and groups to conditions on their spaces of harmonic functions. In particular: we show that a group with a finitely supported symmetric measure has a finite-dimensional space of harmonic functions if and only if it is virtually cyclic; we present a new proof of a result of V. Trofimov that an infinite vertex-transitive graph admits a non-constant harmonic function; we give a new proof of a result of T. Ceccherini-Silberstein, M. Coornaert and J. Dodziuk that the Laplacian on an infinite, connected, locally finite graph is surjective; and we show that the positive harmonic functions on a non-virtually nilpotent linear group span an infinite-dimensional space. Contents2010 Mathematics Subject Classification. 20F65 (primary), 60G50 (secondary).

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“…Do other groups (of non-polynomial volume growth) admit such a "forbidden gap" in the growth of non-constant harmonic functions? See [1,10,18] for precise results in the case of polynomial growth.…”

Section: Open Questionsmentioning

confidence: 99%

Minimal growth harmonic functions on lamplighter groups

Benjamini,

Duminil-Copin,

Kozma

et al. 2016

Preprint

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Polynomials and harmonic functions on discrete groups

Cited by 3 publications

References 0 publications

Harmonic functions of linear growth on solvable groups

Harmonic functions of linear growth on solvable groups

Characterisations of algebraic properties of groups in terms of harmonic functions

Minimal growth harmonic functions on lamplighter groups

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