## Notes of Gady Kozma’s lecture

Harmonic functions of minimal growth

With Amir, Benjamini, Duminil-Copin, Meyerovich, Yadin.

1. Motivation

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 ?

1.1. Examples

What about Grigorchuk group ? Assume ${V=\{Lipschitz harmonic functions\}}$ is finite dimensional. This is a torsion group. Its image in the linear group ${GL(V)}$ is finite. This easily leads to a contradiction. We conclude that, for Grigorchuk group, ${V}$ is infinite dimensional.

What about lamplighter group ? I construct an infinite dimensional space of positive Lipschitz harmonic functions on ${{\mathbb Z}_2 \wr {\mathbb Z}}$. Key : function must be big only in a small part of the unit sphere around each point. Let ${T_r}$ be the stopping time when the lamplighter reaches ${\pm r}$. Consider

$\displaystyle \begin{array}{rcl} h(g,r)=\mathop{\mathbb P}^g(X_{T_r}\textrm{ is a configuration with all negative lamps off}). \end{array}$

This is essentially the probability that the walk of the lamplighter stays in ${{\mathbb Z}_+}$. So this is ${O(1/r)}$, and ${rh(g,r)}$ converges as ${r}$ tends to infinity. There is one such function for each ${g\in G}$ and they are linearly independant.

The lamplighter group ${G_1}$ has no sublinear harmonic functions.

$\displaystyle \begin{array}{rcl} h(g)=\mathop{\mathbb E}_g(h(X_{T_r})). \end{array}$

Sublinearity implies that ${\mathop{\mathbb E}_g(h(X_{T_r})1_{X_{T_r}=r})}$ goes to 0. On the other hand,

$\displaystyle \mathop{\mathbb E}_g(h(X_{T_r})1_{X_{T_r}=0})=h(e),$

showing that ${h(g)=h(e)}$.

2. Result

Theorem 1 Assume ${V=\{Lipschitz harmonic functions\}}$ is finite dimensional. Then ${G}$ has no sub-linear harmonic functions.

2.1. Proof

Assume ${G}$ is amenable. Let ${H}$ be the image in ${GL(V}$ mod constants${)}$. By Tits, we know that ${H}$ is virtually solvable. By Malcev, ${H}$ is virually uniformly triangularizable. Furthermore, ${H}$ preserves a norm (Lipschitz constant of harmonic functions), so it is virtually abelian. Thus ${G}$ has a finite index subgroup ${A}$ such that, for ${h\in V}$ in a suitable basis, ${gh=\lambda h+c}$. Taking commutators shows that ${\lambda=1}$. If ${h}$ is sub-linear, then ${c=0}$. Argument ends with the fact that virtually abelian groups have no sublinear harmonic functions.

2.2. More

Conjecture: ${G}$ has no sublinear harmonic functions if and only if ${G}$ is diffusive, i.e. simple random walk escapes at speed ${\sqrt{n}}$.