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}


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