** Poisson boundary and reduced cohomology **

**1. Motivation **

In 1936, Koebe proved that every planar graph is the incidence graph of a disk packing in . In higher dimensions, obstructions are to be expected. For instance,

Theorem 1 (Benjamini-Schramm)If is the incidence graph of a sphere packing in , then either is -parabolic, or has a non constant -harmonic function.

This motivates us to find properties which imply vanishing of reduced cohomology.

**2. Definitions **

Let be a graph. Denote by the set of functions with gradient in . Then reduced cohomology is

The Poisson boundary of is the space whose bounded functions parametrize bounded harmonic functions on .

Say has , a -dimensional isoperimetric profile, if for every finite set of vertices,

where is the set of edges joining to its complement.

For instance, Varopoulos showed that the Cayley graph of a finitely generated group does not have

**3. Result **

Theorem 2Assume that has . Then, there exists a linear mapdefined for , mapping constants to constants, bounded functions to bounded functions, to for all , and such that

Corollary 3If is amenable, its Cayley graph satisfies for all .

Lemma 4 (Holopainen-Soardi)

Groups without non constant harmonic functions are called Liouville. They are all amenable. This class includes polycyclic groups, groups of intermediate growth, lamplighter over , lamplighter over with a Liouville group of lamps.

On the other hand, lamplighter over is not Liouville

Corollary 5If is Liouville, then for all .

This fails for . Indeed, is the set of functions on the ends mod constants.

**4. Proof **

Let denote the probability to reach in steps from . For a function on vertices, let

The punchline is to prove that

exists pointwise. Estimate

provided .

Definition 6Given finitely supported probability measures , , call a transport pattern from to if .

For instance, is any oriented path from to .

One could try

Now is obtained by diffusing one step, so

Varopoulos tells us that implies the following bound

We want to get a bound on . This yields

This implies pointwise convergence.

Other properties come rather easily. Here is how the last one follows.

Assume . Pick a large -ball containing almost all of ‘s energy and such that the tail of is small. Outside that ball, is small, is close to , so is small. Conclude with the following lemma.

Lemma 7If and tends to 0 at infinity, then in .

Proof. Truncate at , get with finite support. Then pointwise and converges to , so converges to in .