On polynomial vs. super-polynomial growth for finitely
generated groups and harmonic functions
I will discuss results and open problems about harmonic functions for
finitely generated groups, with an emphasis on polynomial vs. super-polynomial growth. By
"polynomial growth" I am implying to at least two different notions:
1. The volume growth of balls in the group (with respect to the word metric).
2. The growth of the sup norm of a harmonic function on a ball.
The investigation is partly motivated by Kleiner’s proof for Gromov’s
theorem on groups of polynomial growth, by Ozawa's more recent proof of Gromov's theorem and by some related
conjectures and hypothetical future applications to problems in geometric group theory.
Most of the results I will present in this talk will be at least a few
years old and based on joint work with Ariel Yadin and Perl, Tointon and Yadin.