## Notes of Romain Tessera’s Lille lecture nr 2

1. Consequence

Corollary 1 Let ${G}$ be a real Lie group. Then ${G}$ is hyperbolic iff it acts isometrically and transitively on a negatively curved Riemannian manifold. Let ${G}$ be a ${p}$-adic Lie group. Then ${G}$ is hyperbolic iff it acts isometrically and transitively on a regular tree.

2. Proof of classification theorem

On shows that unless ${B}$ is Heintze, reduced ${\ell^p}$ cohomology vanishes.

Splits into two cases, whether ${B}$ is unimodular or not.

I will illustrate the argument withe the groups ${Sol(\lambda_1,\lambda_2)}$, semi-direct product of ${{\mathbb R}^2}$ by ${{\mathbb Z}}$ acting via a diagonal matrix with eigenvalues ${\lambda_1,\lambda_2}$.

2.1. Case ${B}$ is non unimodular

Let ${T}$ be the generator of the ${{\mathbb Z}}$ factor. Let ${\Delta}$ be the modular function.

I prove that ${\bar{H}^1_p(Sol(\lambda_1,\lambda_2))=0}$ if ${\lambda_1+\lambda_2\not=0}$.

Lemma 2 Let ${u\in D^p(G)}$. There exists ${u_\infty\in D^p(G)}$ such that ${u-u_\infty\in\ell^p}$. Conversely, if ${u_\infty}$ is constant, cohomology class is 0.

For that example,

$\displaystyle \begin{array}{rcl} txt^{-1}=\begin{pmatrix} e^{\lambda_1}&0\\ 0&e^{-\lambda_2} \end{pmatrix}. \end{array}$

Let ${W=\{g\,;\, t^{-n}wt^n}$ is bounded independantly of ${n\}}$. This a subgroup of the form ${W=(0\times {\mathbb R})\times{\mathbb Z}}$. I show that function ${u}$ is left ${W}$-invariant. Thus ${u}$ cannot have finite energy, unless ${u=0}$.

2.2. Case ${B}$ is unimodular

E.g. ${G=SOL}$, ${\lambda_1+\lambda_2=0}$.

We show that any ${u\in D^p}$ can be approximated by ${L^p}$ functions. We need good F\o lner sequences. Let

$\displaystyle \begin{array}{rcl} F_n=\{(x,y,m)\,;\,|x|\leq e^{2n},\,|y|\leq e^{2n},\,|m|\leq n\}. \end{array}$

One checks that

$\displaystyle \begin{array}{rcl} \frac{|sF_n\Delta F_n|}{|F_n|}\leq \frac{C}{n},\quad\textrm{and}\quad F_n\subset B(100n). \end{array}$

We shall approximate ${u}$ with

$\displaystyle \begin{array}{rcl} u_n(g)=\frac{1}{|F_n|}\int_{F_n}(u(gh)-u(h))\,dh. \end{array}$

${u_n}$ is an average of left translates of expressions ${\rho_h(u)-u}$, so it is in ${\ell^p}$.

The error term ${v_n=u_n-u}$ is the average of ${u}$ over ${F_n}$. Since ${F_n}$ is nearly left invariant, ${v_n}$ is nearly right-invariant, meaning that ${\|\rho_s(v_n)-v_n\|_p}$ tends to 0. This means that ${v_n}$ tends to 0 in ${D^p(G)}$.

2.3. Final step

Let ${G}$ be a locally compact compactly generated group. Let ${S\subset G}$ be open. Let ${G}$ act in a mixing manner on an infinite measure space ${(X,\mu)}$ (e.g. left action of ${G}$ on itself). Then for ever 1-cocycle ${b}$ for the representation ${L^p(X,\mu)}$,

$\displaystyle \begin{array}{rcl} \|b(g)\|_p=o(|g|_S). \end{array}$