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).
We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most k is finite dimensional, to the settings of locally compact groups equipped with measures with non-compact support.
We study a model of growing population that competes for resources. At each
time step, all existing particles reproduce and the offspring randomly move to
neighboring sites. Then at any site with more than one offspring, the particles
are annihilated. This is a nonmonotone model, which makes the analysis more
difficult. We consider the extinction window of this model in the finite
mean-field case, where there are $n$ sites but movement is allowed to any site
(the complete graph). We show that although the system survives for exponential
time, the extinction window is logarithmic.Comment: Published at http://dx.doi.org/10.1214/14-AAP1069 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
We study the connection between the dimension of certain spaces of harmonic functions on a group and its geometric and algebraic properties.Our main result shows that (for sufficiently "nice" random walk measures) a connected, compactly generated, locally compact group has polynomial volume growth if and only if the space of linear growth harmonic functions has finite dimension. This characterization is interesting in light of the fact that Gromov's theorem regarding finitely generated groups of polynomial growth does not have an analog in the connected case. That is, there are examples of connected groups of polynomial growth that are not nilpotent by compact. Also, the analogous result for the discrete case has only been established for solvable groups, and is still open for general finitely generated groups.
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