** Proper isometric actions of hyperbolic groups on spaces **

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

Theorem 1 (Yu 2005)For every hyperbolic group , for large enough, acts properly on .

We build upon Yu’s work and get

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

Theorem 3 (Emerson-Nica 2011)Let be a non-elementary hyperbolic group. Then, for large enough, acts properly on .

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

Theorem 4 (Bourdon 2011)Let be a non-elementary hyperbolic group. Then, for large enough, acts properly on .

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

**1. Free groups **

I describe a (overly) complicated action of on , which will generalize.

Let , use the obvious measure on the , on . Define a cocycle over the action on by

where is Gromov product based at identity.

Then

**2. Möbius actions **

Let be a compact metric space. Say a bi-Lipschitz map satisfies GMV (geometric mean value property) if, for all , ,

Here, denotes the pointwise Lipschitz constant at .

This holds if and only if is *Möbius*, i.e. cross-ratio preserving.

If has Hausdorff dimension with Hausdorff -measure , the measure

is Möbius invariant.

Note that

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

is a cocycle for the action of the Möbius group on .

Lemma 5If is Ahlfors regular, is in for all .

Thus we get an affine action of Möb on .

**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 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 on such that– is -invariant,

– is roughly geodesic,

– is quasi-isometric to word metric,

– the corresponding visual metric extends to the boundary,

– is a metric on the boundary, provided ,

– acts on by Möbius transformations

Proposition 7 (Blachère-Haïssinsky-Mathieu 2010)is Ahlfors regular.

We get an expression for the invariant measure on , a cocycle and an affine isometric action on , for all , Mineyev’s hyperbolic dimension.

Proposition 8tends to infinity as tends to infinity.

**4. Link with -cohomology **

Theorem 9

is a proper cocycle for the linear action of on .