## Notes of Marc Bourdon’s lecture

Some applications of ${\ell_p}$-cohomology to the boundaries of Gromov hyperbolic spaces

Joint work with Bruce Kleiner.

1. ${\ell_p}$ cohomology

1.1. Definition

Definition 1 Let ${X}$ be a contractible hyperbolic simplicial complex with bounded geometry. Its ${\ell_p}$-cohomology is the following quotient space,

$\displaystyle \begin{array}{rcl} \ell_p H^1(X)=\{f:X^{(0)}\rightarrow{\mathbb R}\,;\,df\in\ell_p(X^{(1)})\}/(\ell_p(X^{(0)})+{\mathbb R}). \end{array}$

Theorem 2 (Strichartz, Bourdon-Pajot) If ${df\in\ell_p(X^{(1)})}$, then ${f}$ admits a measurable radial extension ${f_{\infty}:\partial X\rightarrow{\mathbb R}}$. Furthermore, the map ${f\mapsto f_{\infty}}$ mod constants descends to an injective map on ${\ell_p}$ cohomology.

Radial means that ${f_{\infty}(\xi)}$ is the limit of ${f}$ along rays representing ${\xi}$.

1.2. Numerical invariants

It is convenient to stick to continuous functions arising as radial extensions. These form an algebra of functions, from which further invariants can be extracted.

Definition 3

$\displaystyle \begin{array}{rcl} \ell_p H^1_{cont}(X)=\{[f]\in\ell_pH^{1}(X)\,;\, f_{\infty}\textrm{ is continuous}\}. \end{array}$

The ${\ell_p}$ equivalence relation on ${\partial X}$ is defined by

$\displaystyle \begin{array}{rcl} z\sim z' \Leftrightarrow \forall [f]\in\ell_p H^1_{cont}(X),\,f(z)=f(z'). \end{array}$

Define numerical invariants

$\displaystyle \begin{array}{rcl} p_{\not=0}(X)&=&\inf\{p\geq 1\,;\,\ell_p H^1_{cont}(X)\not=0\}=\inf\{p\geq 1\,;\,\partial X_{/\sim} =*\},\\ p_{sep}(X)&=&\inf\{p\geq 1\,;\,\ell_p H^1_{cont}(X)\textrm{ separates points }\}=\inf\{p\geq 1\,;\,\partial X_{/\sim} =\partial X\}. \end{array}$

Theorem 4 (Kleiner, Carrasco) Assume ${Isom(X)}$ is cocompact. Then

$\displaystyle \begin{array}{rcl} p_{sep}(X)=\mathrm{dim}_{AR}(\partial X). \end{array}$

Theorem 5 (Bourdon-Pajot) Assume ${\partial\Gamma}$ satisfies the combinatorial ${Q}$-Loewner property for some ${Q}$. Then

$\displaystyle \begin{array}{rcl} p_{sep}(X)=p_{\not=}(X)=Q. \end{array}$

I will not define combinatorial Loewner property. It is quasi-Möbius invariant. It follows from the usual, analytic, Loewner property. The converse is conjectured to hold (Kleiner).

Example 1 Boundaries if rank one symmetric spaces, of Fuchsian buildings, are Loewner. Poincaré inequality implies Loewner.

2. New results

2.1. Ways to construct ${\ell_p}$ cohomology classes

The classical procedure is

Proposition 6 (Elek, Pansu) If ${p>\mathrm{HausDim}(\partial X)}$, then ${Lip(\partial X)\subset\ell_p H^1_{cont}(X)}$.

I will use relative classes. Assume ${Y\subset X}$ is a subcomplex. Then, by excision,

$\displaystyle \begin{array}{rcl} \ell_pH^2(X,X\setminus Y)=\ell_pH^1(Y,Front(Y)), \end{array}$

where ${Front(Y)=Y\setminus\dot{Y}}$ denotes the boundary of ${Y}$ as a subset of ${Y}$.

I will illustrate this construction in two situations.

2.2. Amalgamated products

Let ${A}$, ${B}$ be hyperbolic groups, let ${C}$ be a quasi-convex and malnormal subgroup both of ${A}$ and of ${B}$. Let ${\Gamma=A*_C B}$. Take as ${X}$ (resp. ${Y}$) the Cayley graph of ${\Gamma}$ (resp. ${A}$). Then ${Front(Y)}$ is a disjoint union of copies of the Cayley graph of ${C}$, indexed by ${A/C}$.

Theorem 7 Assume ${C}$ is separable in ${A}$, i.e. there exist a sequence ${A_n}$ of finite index subgroups of ${A}$ such that ${\bigcap A_n=C}$. Let ${\Gamma_n =A_n *_C B}$. Then

1. If ${p_{sep}(A), then for ${n}$ large enough,

$\displaystyle \begin{array}{rcl} p_{\not=0}(\Gamma_n)

2. If ${p_{sep}(A), then, for ${p\in(p_{sep}(A),p_{\not=0}(B))}$ and for ${n}$ large enough, the ${\ell_p}$ equivalence relation on ${\partial\Gamma_n}$ is of the form

$\displaystyle \begin{array}{rcl} z\sim z' \Leftrightarrow z=z' \textrm{ or }\exists g\in\Gamma_n \textrm{ such that }\{z,z'\}\subset g\partial B. \end{array}$

In other words, when replacing ${A}$ with ${A_n}$, the cosets ${aC}$ get far away from each other, and one can construct functions which are constant on these cosets, while keeping their gradients in ${\ell_p}$.

Example 2 Let ${A}$ and ${B}$ have Sierpinski carpet boundaries with different conformal dimensions. Amalgamate them along peripheral Fuchsian subgroups. The result cannot have Loewner property.

2.3. Polygonal complexes

Theorem 8 Let ${X}$ be a simply connexted 2-complex such that

1. Every 2-cell is isometric to an Euclidean polygon with at least ${n\geq 7}$ sides.
2. Every pair of 2-cells meet along at most a vertex or an edge.
3. Every edge belongs to at most ${k\geq 2}$ 2-cells.

(thus ${X}$ is hyperbolic). Then

$\displaystyle \begin{array}{rcl} p_{sep}(X)\leq 1+\frac{\log(k-1)}{\log(n-5)}. \end{array}$

In many cases, this upper bound is much better than previously known bounds (independent on ${n}$), based on area growth for the natural ${CAT(-1)}$ metric.

For this, one first constructs a tree in ${X}$ using geodesic segments in cells orthogonal to edges. Let ${Y}$ be a thickening of that tree. The 2-cells in ${Y}$ are polygons with two types of edges. It is not hard to compute when the relative ${\ell_p}$ cohomology of ${Y}$ separates ends. Indeed, push frontier edges to infinity, replacing each cell by an ideal triangle. The result, like a hyperbolic building, admits retractions to hyperbolic plane. Pull back a smooth function on the closed disk. This yields a function which is constant on frontier edges, with controllable ${\parallel df\parallel_p}$.

This construction seems specific to dimension 2.

Remark 1 ${p_{sep}}$ increases under quasi-isometric embeddings, but ${p_{\not=0}}$ does not.