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.
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.
is a cocycle for the action of the Möbius group on , and
is a cocycle for the action of the Möbius group on .
Lemma 5 If 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 8 tends to infinity as tends to infinity.
4. Link with -cohomology
is a proper cocycle for the linear action of on .