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}}.

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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