Notes of Bogdan Nica’s lecture

Proper isometric actions of hyperbolic groups on {L^p} spaces

As far as affine isometric actions on {L^2} spaces, hyperbolic groups may have Kazhdan’s property, Haagerup property, or neither. As far as affine isometric actions on {L^p} spaces, Yu’s theorem asserts that

Theorem 1 (Yu 2005) For every hyperbolic group {\Gamma}, for {p} large enough, {\Gamma} acts properly on {L^p(\Gamma \times \Gamma)}.

We build upon Yu’s work and get

Theorem 2 (Emerson-Nica 2010) Let {\Gamma} be a non-elementary hyperbolic group with boundary {\partial\Gamma}. There exist (meaningful) finitely summable Fredholm modules for cross-product {C(\partial\Gamma )\times\Gamma}.

Theorem 3 (Emerson-Nica 2011) Let {\Gamma} be a non-elementary hyperbolic group. Then, for {p} large enough, {\Gamma} acts properly on {L^p(\partial\Gamma \times \partial\Gamma )}.

Our bound on {p} depends on what Mineyev calls hyperbolic dimension. Good to compare with

Theorem 4 (Bourdon 2011) Let {\Gamma} be a non-elementary hyperbolic group. Then, for {p} large enough, {\Gamma} acts properly on {L^p(\Gamma \times\cdots\times \Gamma)}.

Bourdon’s bound is the Ahlfors-regular conformal dimension.

1. Free groups

I describe a (overly) complicated action of {F_n} on {L^p}, which will generalize.

Let {g=2n-1}, use the obvious measure on the {\partial\Gamma}, on {\partial\Gamma \times \partial\Gamma}. Define a cocycle {c} over the {F_n} action on {\partial\Gamma \times \partial\Gamma} by

\displaystyle  \begin{array}{rcl}  c_g(\xi,\omega) = (g,\xi)-(g,\omega), \end{array}

where {(g,\xi)} is Gromov product based at identity.

Then

\displaystyle  \begin{array}{rcl}  | c_g |_{L^p} \sim | g |^{1/p} . \end{array}

2. Möbius actions

Let {X} be a compact metric space. Say a bi-Lipschitz map {g : X \rightarrow X} satisfies GMV (geometric mean value property) if, for all {x}, {y \in X},

\displaystyle  \begin{array}{rcl}  d(gx,gy)^2 = |g'(x)| |g'(y)| d(x,y)^2 . \end{array}

Here, {|g'(x)|} denotes the pointwise Lipschitz constant at {x}.

This holds if and only if {g} is Möbius, i.e. cross-ratio preserving.

If {X} has Hausdorff dimension {D} with Hausdorff {D}-measure {\mu_D}, the measure

\displaystyle  \begin{array}{rcl}  \nu = d^{-2D} d\mu_D(x) d\mu_D(y) \end{array}

is Möbius invariant.

Note that

\displaystyle  \begin{array}{rcl}  g \rightarrow \log | (g^{-1})' | \end{array}

is a cocycle for the action of the Möbius group on {X}, and

\displaystyle  \begin{array}{rcl}  g \rightarrow c_g(x,y) = \log | (g^{-1})'(x) | - \log | (g^{-1})'(y) | \end{array}

is a cocycle for the action of the Möbius group on {X \times X}.

Lemma 5 If {X} is Ahlfors regular, {c_g} is in {L^p} for all {p>D}.

Thus we get an affine action of Möb{(X)} on {L^p(X \times X,\nu)}.

3. Hyperbolic groups

We need a metric on the boundary.

Coornaert (1993): Take the word metric and the corresponding visual metric. This is Ahlfors regular, but {\Gamma} action is only quasi-Möbius.

In his work on Baum-Connes, Mineyev introduced the hat metric.

Proposition 6 (Mineyev 2005/2007) There exists a new metric {\hat{d}} on {\Gamma} such that

{\hat{d}} is {\Gamma}-invariant,

{\hat{d}} is roughly geodesic,

{\hat{d}} is quasi-isometric to word metric,

– the corresponding visual metric extends to the boundary,

{\hat{d}_\epsilon = \exp(-\epsilon <,>)} is a metric on the boundary, provided {\epsilon \leq 1},

{\Gamma} acts on {(\partial\Gamma ,\hat{d}_\epsilon )} by Möbius transformations

Proposition 7 (Blachère-Haïssinsky-Mathieu 2010) {\hat{d}_\epsilon} is Ahlfors regular.

We get an expression for the invariant measure on {\partial\Gamma \times \partial\Gamma}, a cocycle and an affine isometric action on {L^p(\partial\Gamma \times \partial\Gamma )}, for all {p > D}, Mineyev’s hyperbolic dimension.

Proposition 8 {| c_g |_{L^p}} tends to infinity as {| g |} tends to infinity.

4. Link with {L^p}-cohomology

Theorem 9

\displaystyle  \begin{array}{rcl}  g \rightarrow <g,.> \end{array}

is a proper cocycle for the linear action of {\Gamma} on {H_p^1(\Gamma)}.

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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 http://www.math.ens.fr/metric2011/
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