## Notes of Matias Carrasco-Piaggio’s lecture

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 ${X_\alpha={\mathbb R}\times N}$ where ${N}$ is nilpotent, the ${{\mathbb R}}$ action is generated by a derivation ${\alpha}$ all of whose eigenvalues have positive real parts.

1.2. The pointed sphere conjecture

${\partial X_\alpha}$ is a sphere on which ${N}$ has two orbits, one of them a fixed point, ${\infty}$.

Conjecture (Cornulier’s terminology): ${QIsom(X_\alpha)}$ fixes the point ${\infty}$ unless ${X_\alpha}$ is isometric to a rank 1 symmetric space.

1.3. Known results

1. Up to qi, one can assume that ${X_\alpha}$ is purely real, i.e. the eigenvalues ${\mu_1\leq\cdots\leq\mu_n}$ of ${\alpha}$ are real (Cornulier).
2. Say ${X_\alpha}$ is of Carnot type if

$\displaystyle \begin{array}{rcl} LieSpan(ker(\alpha-\mu_1))=\mathfrak{n}. \end{array}$

Pansu showed that if ${\alpha}$ is diagonalisable and ${X_\alpha}$ is not of Carnot type, then PSC holds.

3. Xie showed that PSC holds if ${N}$ is abelian or if ${N}$ is Heisenberg group and ${\alpha}$ 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 ${X_\alpha}$‘s which are purely real and not of Carnot type.

In other words, we treat the case where ${\alpha}$ is not diagonalisable.

2.1. Scheme of proof

The idea is that ${\ell^p}$ cohomology in a neighborhood of a point of the ideal boundary detects ${\infty}$. I illustrate this with ${N={\mathbb R}^2}$ and one of the following derivations,

$\displaystyle \begin{array}{rcl} (i)\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix},\quad (ii)\begin{pmatrix} 1&0 \\ 0&2 \end{pmatrix},\quad (iii)\begin{pmatrix} 1&1 \\ 0&1 \end{pmatrix}. \end{array}$

Pull-back functions from the “Heintze cone” ${Z_\alpha={\mathbb Z}^2\setminus X_\alpha}$. In case (ii), ${\ell^pH^1(Z_\alpha)\not=0}$ starting from ${p=3/2}$. Pulled-back functions are ${{\mathbb Z}^2}$-perodic, they cannot have finite energy but their energy is finite locally near points of the boundary different from ${\infty}$. This cannot happen near ${\infty}$, showing how ${\infty}$ differs quasiisometrically from other points.

In case (iii), ${\ell^pH^1(Z_\alpha)\not=0}$ starting from ${p=2}$ only. For ${p>2}$, ${\ell^pH^1(X_\alpha)\not=0}$, achieved for instance by functions which may vanish in a neighborhood of point ${\infty}$, the method collapses.

2.2. Case (iii), non diagonalizable derivation

The metrics differs from hyperbolic metric (case (i)) only logarithmically. ${\ell^p}$ 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

$\displaystyle \begin{array}{rcl} \phi(x)=\frac{|x|^p}{\log(e+\frac{1}{|x|})^{\kappa}}. \end{array}$

In example (iii), we see that for ${p=2}$, for ${\kappa>3}$, there is a lot of ${\ell^\phi}$ cohomology. For ${1<\kappa<3}$, cohomology consists of functions depending on ${y}$ only. Thus in this regime, ${\ell^\phi}$ cohomology localized near a point of the boundary changes at ${\infty}$.

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.

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 ?