## 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

$\displaystyle \begin{array}{rcl} g \rightarrow \end{array}$

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