Polynomial growth harmonic functions on groups of polynomial volume growth

Hua, Bobo; Jost, Jürgen

doi:10.1007/s00209-015-1436-5

Cited by 6 publications

(2 citation statements)

References 54 publications

(121 reference statements)

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“…More generally, Theorem 1.6 recovers and refines another relatively recent result of Hua & Jost [10] stating that when G has a finite-index nilpotent subgroup N we have…”

Section: Introductionsupporting

confidence: 75%

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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“…More generally, Theorem 1.6 recovers and refines another relatively recent result of Hua & Jost [10] stating that when G has a finite-index nilpotent subgroup N we have…”

Section: Introductionsupporting

confidence: 75%

Polynomials and harmonic functions on discrete groups

Meyerovitch

Perl

Tointon

et al. 2016

Trans. Amer. Math. Soc.

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“…It seems hard to calculate the precise dimension of polynomial growth harmonic functions on groups of polynomial volume growth. One step back, we give a dimension estimate by the geometric method in [26].…”

Section: Theorem 14 (Dimension Calculation)mentioning

confidence: 99%

Polynomial growth harmonic functions on finitely generated abelian groups

2013

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In the present paper, we develop geometric analysis techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We provide a geometric analysis proof of the classical Heilbronn theorem [22] and the recent Nayar theorem [44] on polynomial growth harmonic functions on lattices Z n that does not use a representation formula for harmonic functions. In the abelian group case, by Yau's gradient estimate we actually give a simplified proof of a general polynomial growth harmonic function theorem of Alexopoulos [3]. We calculate the precise dimension of the space of polynomial growth harmonic functions on finitely generated abelian groups by linear algebra, rather than by Floquet theory Kuchment-Pinchover [30]. While the Cayley graph not only depends on the abelian group, but also on the choice of a generating set, we find that this dimension depends only on the group itself. Moreover, we also calculate the dimension of solutions to higher order Laplace operators.

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Equivariant discretizations of diffusions and harmonic functions of bounded growth

Ballmann

Polymerakis

2022

Isr. J. Math.

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Polynomial growth harmonic functions on groups of polynomial volume growth

Cited by 6 publications

References 54 publications

Polynomials and harmonic functions on discrete groups

Polynomials and harmonic functions on discrete groups

Polynomial growth harmonic functions on finitely generated abelian groups

Equivariant discretizations of diffusions and harmonic functions of bounded growth

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