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 ?

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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