Orlicz spaces and quasiisometries of Heintze groups
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 1 PSC 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 2 Conformal dimension of boundary is not attained for case (iii).
This was proved earlier by Haissinsky-Pilgrim, Bonk…, using conformal dynamics.
Is it known that Carnot type and non Carnot type cannot be quasiisometric ?
Does Orlicz cohomology help understanding quasiisometries of nilpotent groups ?