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Register a new userPolynomial Growth Harmonic Functions on Finitely Generated Abelian Groups
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In the present paper, we develop geometric analytic techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We develop a geometric analytic proof of the classical Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic functions on lattices $mathds{Z}^n$ that does not use a representation formula for harmonic functions. We also calculate the precise dimension of the space of polynomial growth harmonic functions on finitely generated abelian groups. 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.
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We consider harmonic functions of polynomial growth of some order $d$ on Cayley graphs of groups of polynomial volume growth of order $D$ w.r.t. the word metric and prove the optimal estimate for the dimension of the space of such harmonic functions. More precisely, the dimension of this space of harmonic functions is at most of order $d^{D-1}$. As in the already known Riemannian case, this estimate is polynomial in the growth degree. More generally, our techniques also apply to graphs roughly isometric to Cayley graphs of groups of polynomial volume growth.
We study the frog model on Cayley graphs of groups with polynomial growth rate $D geq 3$. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one of these particles is active when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices. We prove that the activation time of particles grows at least linearly and we show that in the abelian case with any finite generator set the set of activated sites has a limiting shape.
For an infinite penny graph, we study the finite-dimensional property for the space of harmonic functions, or ancient solutions of the heat equation, of polynomial growth. We prove the asymptotically sharp dimensional estimate for the above spaces.
A unital $ell$-group $(G,u)$ is an abelian group $G$ equipped with a translation-invariant lattice-order and a distinguished element $u$, called order-unit, whose positive integer multiples eventually dominate each element of $G$. We classify finitely generated unital $ell$-groups by sequences $mathcal W = (W_{0},W_{1},...)$ of weighted abstract simplicial complexes, where $W_{t+1}$ is obtained from $W_{t}$ either by the classical Alexander binary stellar operation, or by deleting a maximal simplex of $W_{t}$. A simple criterion is given to recognize when two such sequences classify isomorphic unital $ell$-groups. Many properties of the unital $ell$-group $(G,u)$ can be directly read off from its associated sequence: for instance, the properties of being totally ordered, archimedean, finitely presented, simplicial, free.
Suppose $(M,g)$ is a Riemannian manifold having dimension $n$, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity $C(X)$ is an Euclidean cone over the cross-section $X$. Denote by $alpha=lim_{rrightarrowinfty}frac{mathrm{Vol}(B_{r}(p))}{r^{n}}$ the asymptotic volume ratio. Let $h_{k}=h_{k}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $k$. In this paper, we prove a upper bound of $h_{k}$ in terms of the counting function of eigenvalues of $X$. As a corollary, we obtain $lim_{krightarrowinfty}k^{1-n}h_{k}=frac{2alpha}{(n-1)!omega_{n}}$. These results are sharp, as they recover the corresponding well-known properties of $h_{k}(mathbb{R}^{n})$. In particular, these results hold on manifolds with nonnegative sectional curvature and maximal volume growth.
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