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 ,