Harmonic functions of minimal growth
With Amir, Benjamini, Duminil-Copin, Meyerovich, Yadin.
Gromov’s polynomial growth theorem. Kleiner’s new proof that uses Lipschitz harmonic functions.
Any group has non trivial Lipschitz harmonic functions. Polynomial growth implies that they form a finite dimensional space (inspired by a similar result by Colding and Minicozzi on Riemannian manifolds).
Question. Conversely, does finite dimensionality of the space of Lipschitz harmonic functions imply polynomial growth ?
What about Grigorchuk group ? Assume is finite dimensional. This is a torsion group. Its image in the linear group is finite. This easily leads to a contradiction. We conclude that, for Grigorchuk group, is infinite dimensional.
What about lamplighter group ? I construct an infinite dimensional space of positive Lipschitz harmonic functions on . Key : function must be big only in a small part of the unit sphere around each point. Let be the stopping time when the lamplighter reaches . Consider
This is essentially the probability that the walk of the lamplighter stays in . So this is , and converges as tends to infinity. There is one such function for each and they are linearly independant.
The lamplighter group has no sublinear harmonic functions.
Sublinearity implies that goes to 0. On the other hand,
showing that .
Theorem 1 Assume is finite dimensional. Then has no sub-linear harmonic functions.
Assume is amenable. Let be the image in mod constants. By Tits, we know that is virtually solvable. By Malcev, is virually uniformly triangularizable. Furthermore, preserves a norm (Lipschitz constant of harmonic functions), so it is virtually abelian. Thus has a finite index subgroup such that, for in a suitable basis, . Taking commutators shows that . If is sub-linear, then . Argument ends with the fact that virtually abelian groups have no sublinear harmonic functions.
Conjecture: has no sublinear harmonic functions if and only if is diffusive, i.e. simple random walk escapes at speed .