**1. Consequence **

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

**2. Proof of classification theorem **

On shows that unless is Heintze, reduced cohomology vanishes.

Splits into two cases, whether is unimodular or not.

I will illustrate the argument withe the groups , semi-direct product of by acting via a diagonal matrix with eigenvalues .

** 2.1. Case is non unimodular **

Let be the generator of the factor. Let be the modular function.

I prove that if .

**Lemma 2** * Let . There exists such that . Conversely, if is constant, cohomology class is 0. *

For that example,

Let is bounded independantly of . This a subgroup of the form . I show that function is left -invariant. Thus cannot have finite energy, unless .

** 2.2. Case is unimodular **

E.g. , .

We show that any can be approximated by functions. We need good F\o lner sequences. Let

One checks that

We shall approximate with

is an average of left translates of expressions , so it is in .

The error term is the average of over . Since is nearly left invariant, is nearly right-invariant, meaning that tends to 0. This means that tends to 0 in .

** 2.3. Final step **

Let be a locally compact compactly generated group. Let be open. Let act in a mixing manner on an infinite measure space (e.g. left action of on itself). Then for ever 1-cocycle for the representation ,

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## 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/