**1. Formulae **

Free products of infinite groups: add up for , add up plus one.

Free products with amalgamation: idem ( add up…) provided amalgamated subgroup is amenable.

**2. Betti numbers of amenable groups **

Following Cheeger and Gromov, I show that Betti numbers vanish for infinite amenable groups. I make the simplifying assumption that group acts freely and cocompactly on a contractible simplicial complex .

The result obviously follows from

**Lemma 1** * The forgetful map to ordinary cohomology
*

*
** is injective. *

We study the -dimension of the kernel of the forgetful map. By definition,

Let be a fundamental domain. Let be an increasing F\o lner sequence of finite subsets of . Let . Then

The -dimension can be rewritten

Let denote restriction of cochains to . I claim that

Indeed, the composition is a contraction, so . Thus the block of the matrix of , , satisfies

Now I claim that

Finally, for ,

If the support of does not meet , then , and thus and . Thus contains all the kernels of linear froms “evaluation on simplices of ”. This yield the codimension estimate.

Since , we see that

which tends to 0, so and the Lemma is proved.

**3. Euler-Poincaré characteristic **

**Theorem 2 (Atiyah)** *
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*
** *

I will prove a stronger inequality,

** 3.1. Morse inequalities **

**Theorem 3** * For all ,
*

*
** where is the number of -simplices in . *

Proof: usual diagram chasing.

** 3.2. Consequences **

**Corollary 4** * If is finitely generated and infinite, with generators, .*

*
*
If is finitely presented with generators, then

* *

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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/