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Transactions of the American Mathematical Society

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Polynomials and harmonic functions on discrete groups


Authors: Tom Meyerovitch, Idan Perl, Matthew Tointon and Ariel Yadin
Journal: Trans. Amer. Math. Soc. 369 (2017), 2205-2229
MSC (2010): Primary 20F65; Secondary 05C25
DOI: https://doi.org/10.1090/tran/7050
Published electronically: October 27, 2016
MathSciNet review: 3581232
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Abstract: 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.


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Additional Information

Tom Meyerovitch
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Be’er Sheva, Israel
Email: mtom@math.bgu.ac.il

Idan Perl
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Be’er Sheva, Israel
Email: perli@math.bgu.ac.il

Matthew Tointon
Affiliation: Laboratoire de Mathématiques, Université Paris-Sud 11, 91405 Orsay cedex, France
Address at time of publication: Homerton College, University of Cambridge, Cambridge CB2 8PH, United Kingdom
Email: mcht2@cam.ac.uk

Ariel Yadin
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Be’er Sheva, Israel
Email: yadina@math.bgu.ac.il

DOI: https://doi.org/10.1090/tran/7050
Received by editor(s): June 19, 2015
Published electronically: October 27, 2016
Article copyright: © Copyright 2016 American Mathematical Society