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)<p_{sep}(B)}, then for {n} large enough,

    \displaystyle  \begin{array}{rcl}  p_{\not=0}(\Gamma_n)<p_{sep}(\Gamma_n). \end{array}

  2. If {p_{sep}(A)<p_{\not=0}(B)}, 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.


About metric2011

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