## Notes of Antoine Gournay’s lecture

Poisson boundary and reduced ${\ell^p}$ cohomology

1. Motivation

In 1936, Koebe proved that every planar graph is the incidence graph of a disk packing in ${S^2}$. In higher dimensions, obstructions are to be expected. For instance,

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

This motivates us to find properties which imply vanishing of reduced ${\ell^d}$ cohomology.

2. Definitions

Let ${G}$ be a graph. Denote by ${D^p(G)}$ the set of functions with gradient in ${\ell^p}$. Then reduced ${\ell^p}$ cohomology is

$\displaystyle \begin{array}{rcl} \ell^p\bar{H}^1(G)=D^p(G)/\overline{\ell^p(G)+{\mathbb R}}. \end{array}$

The Poisson boundary of ${G}$ is the space whose bounded functions parametrize bounded harmonic functions on ${G}$.

Say ${G}$ has ${IS_d}$, a ${d}$-dimensional isoperimetric profile, if for every finite set ${F}$ of vertices,

$\displaystyle \begin{array}{rcl} |F|^{d-1/d}\leq K\,|\partial F|, \end{array}$

where ${\partial F}$ is the set of edges joining ${F}$ to its complement.

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

3. Result

Theorem 2 Assume that ${G}$ has ${IS_d}$. Then, there exists a linear map

$\displaystyle \begin{array}{rcl} \pi:D^p(G)\rightarrow \mathcal{H}(G) \end{array}$

defined for ${p<2d}$, mapping constants to constants, bounded functions to bounded functions, ${D^p}$ to ${D^q}$ for all ${q>\frac{pd}{d-2p}}$, and such that

$\displaystyle \begin{array}{rcl} [f]=0\textrm{ in }\ell^p\bar{H}^1(G)\quad \Leftrightarrow\quad \pi(f)\textrm{ is constant}. \end{array}$

Corollary 3 If ${\Gamma}$ is amenable, its Cayley graph satisfies ${\ell^p\bar{H}^1(G)=0}$ for all ${p\in[1,2]}$.

Lemma 4 (Holopainen-Soardi)

$\displaystyle \begin{array}{rcl} (D^p\cap\ell^{\infty})/\overline{\ell^p+{\mathbb R}}=0\quad \Leftrightarrow\quad D^p/\overline{\ell^p+{\mathbb R}}=0. \end{array}$

Groups without non constant harmonic functions are called Liouville. They are all amenable. This class includes polycyclic groups, groups of intermediate growth, lamplighter over ${{\mathbb Z}^2}$, lamplighter over ${{\mathbb Z}}$ with a Liouville group of lamps.

On the other hand, lamplighter over ${{\mathbb Z}^3}$ is not Liouville

Corollary 5 If ${\Gamma}$ is Liouville, then ${\ell^p\bar{H}^1(G)=0}$ for all ${p>1}$.

This fails for ${p=1}$. Indeed, ${\ell^1\bar{H}^1(G)}$ is the set of functions on the ends mod constants.

4. Proof

Let ${P_x^m(y)}$ denote the probability to reach ${y}$ in ${m}$ steps from ${x}$. For a function ${f}$ on vertices, let

$\displaystyle \begin{array}{rcl} P^n f(x)=\sum_y f(y) P_x^n(y). \end{array}$

The punchline is to prove that

$\displaystyle \begin{array}{rcl} \pi(f)=\lim_{n\rightarrow\infty}P^n(f) \end{array}$

exists pointwise. Estimate

$\displaystyle \begin{array}{rcl} (P_x^{n+m}f-P_x^n f)(x)=\langle f,d^*\tau\rangle=\langle df,\tau\rangle \end{array}$

provided ${d^*\tau=P_x^{n+m}-P_x^n}$.

Definition 6 Given finitely supported probability measures ${\phi}$, ${\xi}$, call ${\tau}$ a transport pattern from ${\xi}$ to ${\phi}$ if ${d^*\tau=\phi-\xi}$.

For instance, ${\tau_{\delta_x,\delta_y}}$ is any oriented path from ${x}$ to ${y}$.

One could try

$\displaystyle \begin{array}{rcl} \tau_{P_x^{n+m},P_x^n}=\sum_{i=0}^{m-1}\tau_{P_x^{n+i+1},P_x^{n+i}}. \end{array}$

Now ${P_x^{n+i+1}}$ is obtained by diffusing ${P_x^{n+i}}$ one step, so

$\displaystyle \begin{array}{rcl} \tau_{P_x^{n+i+1},P_x^{n+i}}\sim P^{n+i}. \end{array}$

Varopoulos tells us that ${IS_d}$ implies the following bound

$\displaystyle \begin{array}{rcl} \sup_x \|P_x^n\|_{\ell^{\infty}}\leq K\, n^{-d/2}. \end{array}$

We want to get a bound on ${\|\sum_i P_x^i\|_{\ell^p}}$. This yields

$\displaystyle \begin{array}{rcl} \sup_x \|P_x^i\|_{\ell^{p'}}\leq K\,n^{-d/2p}. \end{array}$

This implies pointwise convergence.

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

Assume ${\pi(f)=0}$. Pick a large ${n}$-ball containing almost all of ${f}$‘s energy and ${n}$ such that the tail of ${\sum\|P_x^i\|_{\ell^{p'}}}$ is small. Outside that ball, ${P^n f-f}$ is small, ${P^nf}$ is close to ${\pi(f)=0}$, so ${f}$ is small. Conclude with the following lemma.

Lemma 7 If ${f\in D^p}$ and tends to 0 at infinity, then ${[f]=0}$ in ${\ell^p\bar{H}^1(G)}$.

Proof. Truncate ${f}$ at ${t}$, get ${f_t}$ with finite support. Then ${|df_t|\leq |df|}$ pointwise and ${df_t}$ converges to ${df}$, so ${df_t}$ converges to ${df}$ in ${\ell^p}$.