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}.


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
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