Hyperbolic Lie groups
1. Non vanishing of cohomology
Theorem 1 Let 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 .
Gromov showed that non elementary hyperbolic groups contain quasiisometrically embedded trees. The argument (ping-pong) extends to compactly generated groups.
Corollary 2 If is hyperbolic as a metric space, then for large enough.
3. Hyperbolic Lie groups
Question. Which connected Lie groups are hyperbolic ?
Example 1 Heintze 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.
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 .
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.