Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] Georgios K. Alexopoulos, Random walks on discrete groups of polynomial volume growth, Ann. Probab. 30 (2002), no. 2, 723-801. MR 1905856, https://doi.org/10.1214/aop/1023481007
  • [2] H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3) 25 (1972), 603-614. MR 0379672
  • [3] Itai Benjamini, Hugo Duminil-Copin, Gady Kozma, and Ariel Yadin, Disorder, entropy and harmonic functions, Ann. Probab. 43 (2015), no. 5, 2332-2373. MR 3395463, https://doi.org/10.1214/14-AOP934
  • [4] Tobias H. Colding and William P. Minicozzi II, Harmonic functions on manifolds, Ann. of Math. (2) 146 (1997), no. 3, 725-747. MR 1491451, https://doi.org/10.2307/2952459
  • [5] Ben Green and Terence Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2) 175 (2012), no. 2, 465-540. MR 2877065, https://doi.org/10.4007/annals.2012.175.2.2
  • [6] Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53-73. MR 623534
  • [7] Yves Guivarc'h, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333-379 (French). MR 0369608
  • [8] Marshall Hall Jr., The theory of groups, Chelsea Publishing Co., New York, 1976. Reprinting of the 1968 edition. MR 0414669
  • [9] H. A. Heilbronn, On discrete harmonic functions, Proc. Cambridge Philos. Soc. 45 (1949), 194-206. MR 0030051
  • [10] Bobo Hua and Jürgen Jost, Polynomial growth harmonic functions on groups of polynomial volume growth, Math. Z. 280 (2015), no. 1-2, 551-567. MR 3343919, https://doi.org/10.1007/s00209-015-1436-5
  • [11] Bobo Hua, Jürgen Jost, and Xianqing Li-Jost, Polynomial growth harmonic functions on finitely generated abelian groups, Ann. Global Anal. Geom. 44 (2013), no. 4, 417-432. MR 3132083, https://doi.org/10.1007/s10455-013-9374-0
  • [12] V. A. Kaĭmanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), no. 3, 457-490. MR 704539
  • [13] Bruce Kleiner, A new proof of Gromov's theorem on groups of polynomial growth, J. Amer. Math. Soc. 23 (2010), no. 3, 815-829. MR 2629989, https://doi.org/10.1090/S0894-0347-09-00658-4
  • [14] Michel Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 101-190 (French). MR 0088496
  • [15] A. Leibman, Polynomial mappings of groups, Israel J. Math. 129 (2002), 29-60. MR 1910931, https://doi.org/10.1007/BF02773152
  • [16] A. I. Malcev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949), 9-32 (Russian). MR 0028842
  • [17] Tom Meyerovitch and Ariel Yadin, Harmonic functions of linear growth on solvable groups, Israel J. Math. 216 (2016), no. 1, 149-180. MR 3556965, https://doi.org/10.1007/s11856-016-1406-6
  • [18] M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234
  • [19] Derek J. S. Robinson, A course in the theory of groups, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996. MR 1357169
  • [20] Matthew C. H. Tointon, Freiman's theorem in an arbitrary nilpotent group, Proc. Lond. Math. Soc. (3) 109 (2014), no. 2, 318-352. MR 3254927, https://doi.org/10.1112/plms/pdu005
  • [21] Matthew Tointon, Characterizations of algebraic properties of groups in terms of harmonic functions, Groups Geom. Dyn. 10 (2016), no. 3, 1007-1049. MR 3551188, https://doi.org/10.4171/GGD/375

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20F65, 05C25

Retrieve articles in all journals with MSC (2010): 20F65, 05C25


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

American Mathematical Society