** Hyperbolic Lie groups **

**1. Non vanishing of cohomology **

Theorem 1Let be a connected graph with bounded degree, which is hyperbolic. Assume that contains en quasi-isometrically embedded 3-regular tree . Then for lage enough.

To simplify the proof, I will use an extra assumption: Given a base point , such that every point sits at distance at most from a geodesic ray passing through .

is a Cantor set embedded in . Let and be disjoint open sets that both intersect an together cover . Consider the function on

where . Extend radially to a function on . Then for large enough. Indeed, decays exponentially in terms of the distance of edge to . Its restriction belongs to . It pairs non trivially with the cycle supported on which takes value 1 on an edge, on neighboring edges, on the next layer, and so on. Therefore its reduced cohomology class does not vanish.

**2. cohomology of locally compact groups **

Let be locally compact, compactly generated. Fix a compact generating set. This turns into a metric space.

Fact. There exists a connected bounded degree graph which is quasiisometric to .

Indeed, pick a maximal set of disjoint unit balls in , these are the vertices, and put edges when centers are at distance .

Let

Define

Gromov showed that non elementary hyperbolic groups contain quasiisometrically embedded trees. The argument (ping-pong) extends to compactly generated groups.

Corollary 2If is hyperbolic as a metric space, then for large enough.

**3. Hyperbolic Lie groups **

**Question**. Which connected Lie groups are hyperbolic ?

Example 1Heintze groups, the affine group of the line (it is Heintze), the affine group of , , .

Note these groups either act transitively on negatively curved manifolds, or cocompactly on trees. We shall prove that this is general.

** 3.1. Reduction **

Let be a connected Lie group (over or ). There exists a cocompact closed subgroup and a compact normal subgroup such that satisfies

where is simply connected nilpotent.

Note that one really needs and not .

** 3.2. Classification **

Theorem 3 (Cornulier-Tessera)Let be a connected Lie group (over or ). The following are equivalent.

- is hyperbolic.
- is Heintze in the sense that and the action of on is contracting.
- for some .
- for large enough.

We have only partial results for general compactly generated groups: we can treat (with Caprace and Monod) the case when has an amenable cocompact subgroup.

Note that such a classification fails for discrete groups. For instance, has nonzero reduced cohomology for all (it has infinitely many ends) but it is not hyperbolic.