Polynomial Growth Harmonic Functions on Groups of Polynomial Volume Growth

Details Polynomial Growth Harmonic Functions on Groups of Polynomial Volume Growth Polynomial Growth Harmonic Functions on Groups of Polynomial Volume Growth

2012-01-25

arxiv.org/abs/1201.5238v1

Mathematics

Metric Geometry

18pages

Scientific paper

By proving the polynomial growth harmonic function theorem on Cayley graphs of groups of polynomial volume growth, Kleiner gave a new proof of Gromov's celebrated theorem that every finitely generated group of polynomial volume growth is virtually nilpotent. Then Shalom and Tao provided a quantitative version of that. Their dimension estimates of the space of polynomial growth harmonic functions are exponential in the growth degree which is not asymptotically optimal but sufficient for their applications. In this paper, we prove the optimal dimension estimate of the space of polynomial growth harmonic functions on groups of polynomial volume growth analogous to the Riemannian case which is polynomial in the growth degree. We also obtain the optimal dimension estimate for polynomial growth harmonic functions on graphs roughly isometric to Cayley graphs of groups of polynomial volume growth.

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