Poisson boundary and reduced cohomology
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.
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
Theorem 2 Assume that has . Then, there exists a linear map
defined for , mapping constants to constants, bounded functions to bounded functions, to for all , and such that
Corollary 3 If 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 5 If is Liouville, then for all .
This fails for . Indeed, is the set of functions on the ends mod constants.
Let denote the probability to reach in steps from . For a function on vertices, let
The punchline is to prove that
exists pointwise. Estimate
Definition 6 Given 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 7 If and tends to 0 at infinity, then in .
Proof. Truncate at , get with finite support. Then pointwise and converges to , so converges to in .