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
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)
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.
Corollary 4 If is finitely generated and infinite, with generators, .
If is finitely presented with generators, then