** Orlicz spaces and quasiisometries of Heintze groups **

**1. Introduction **

** 1.1. Heintze groups **

In 1974, J. Heintze classified negatively curved Riemannian homogeneous spaces. He showed that they are left-invariant metrics on Lie groups which as semi-direct products where is nilpotent, the action is generated by a derivation all of whose eigenvalues have positive real parts.

** 1.2. The pointed sphere conjecture **

is a sphere on which has two orbits, one of them a fixed point, .

**Conjecture** (Cornulier’s terminology): fixes the point unless is isometric to a rank 1 symmetric space.

** 1.3. Known results **

- Up to qi, one can assume that is purely real, i.e. the eigenvalues of are real (Cornulier).
- Say is of Carnot type if
Pansu showed that if is diagonalisable and is not of Carnot type, then PSC holds.

- Xie showed that PSC holds if is abelian or if is Heisenberg group and has at least 2 different eigenvalues.

We see that the Carnot type case is the hardest one.

**2. The result **

Theorem 1PSC holds for ‘s which are purely real and not of Carnot type.

In other words, we treat the case where is not diagonalisable.

** 2.1. Scheme of proof **

The idea is that cohomology in a neighborhood of a point of the ideal boundary detects . I illustrate this with and one of the following derivations,

Pull-back functions from the “Heintze cone” . In case (ii), starting from . Pulled-back functions are -perodic, they cannot have finite energy but their energy is finite locally near points of the boundary different from . This cannot happen near , showing how differs quasiisometrically from other points.

In case (iii), starting from only. For , , achieved for instance by functions which may vanish in a neighborhood of point , the method collapses.

** 2.2. Case (iii), non diagonalizable derivation **

The metrics differs from hyperbolic metric (case (i)) only logarithmically. cohomology is not sensible to these log term. This is why, following a suggestion of Tessera, we use Orlicz spaces.

**3. Orlicz spaces **

We use Orlicz gauges of the form

In example (iii), we see that for , for , there is a lot of cohomology. For , cohomology consists of functions depending on only. Thus in this regime, cohomology localized near a point of the boundary changes at .

Note that we extract in this way a new numerical quasiisometry invariant.

Corollary 2Conformal dimension of boundary is not attained for case (iii).

This was proved earlier by Haissinsky-Pilgrim, Bonk…, using conformal dynamics.

** 3.1. Questions **

Is it known that Carnot type and non Carnot type cannot be quasiisometric ?

Does Orlicz cohomology help understanding quasiisometries of nilpotent groups ?