## Notes of Gaboriau’s Lille lecture nr 2

1. Formulae

Free products of infinite groups: ${\beta_n}$ add up for ${n\geq 2}$, ${\beta_1}$ add up plus one.

Free products with amalgamation: idem (${\beta_n}$ add up…) provided amalgamated subgroup is amenable.

2. Betti numbers of amenable groups

Following Cheeger and Gromov, I show that ${\ell^2}$ Betti numbers vanish for infinite amenable groups. I make the simplifying assumption that group ${\Gamma}$ acts freely and cocompactly on a contractible simplicial complex ${L}$.

The result obviously follows from

Lemma 1 The forgetful map to ordinary cohomology

$\displaystyle \begin{array}{rcl} \bar{H}^n_{(2)}(L)\rightarrow H^n(L,{\mathbb R}) \end{array}$

is injective.

We study the ${\Gamma}$-dimension of the kernel ${K}$ of the forgetful map. By definition,

$\displaystyle \begin{array}{rcl} dim_{\Gamma}(K)=\sum_{\sigma\subset D}\langle p_K\sigma,\sigma\rangle. \end{array}$

Let ${D\subset L}$ be a fundamental domain. Let ${F_j}$ be an increasing F\o lner sequence of finite subsets of ${\Gamma}$. Let ${X_j=F_j D}$. Then

$\displaystyle \begin{array}{rcl} |\{g\in F_j\,;\, gD\cap X_j\not=\emptyset\}|=o(|F_j|). \end{array}$

The ${\Gamma}$-dimension can be rewritten

$\displaystyle \begin{array}{rcl} dim_{\Gamma}(K)=\frac{1}{|F_j|}\sum_{\sigma\in F_j D}\langle p_K\sigma,\sigma\rangle. \end{array}$

Let ${p_j}$ denote restriction of cochains to ${X_j}$. I claim that

$\displaystyle \begin{array}{rcl} dim_{\Gamma}(K)\leq\frac{1}{|F_j|}\,dim_{{\mathbb R}}(p_j(K)). \end{array}$

Indeed, the composition ${p_j\circ p_K}$ is a contraction, so ${trace(p_j\circ p_K)\leq dim_{\mathbb R}(im(p_j\circ p_K))}$. Thus the ${F_j}$ block of the matrix of ${p_K}$, ${p_j\circ p_K\circ p_j}$, satisfies

$\displaystyle \begin{array}{rcl} trace(p_j\circ p_K\circ p_j)\leq dim_{\mathbb R}(im(p_j\circ p_K)). \end{array}$

Now I claim that

$\displaystyle \begin{array}{rcl} codim_{\mathbb R}(K\cap ker(p_j))\leq |\partial X_j|. \end{array}$

Finally, for ${h\in K}$, ${h=\delta b}$

$\displaystyle \begin{array}{rcl} \|p_j(h)\|^2=\langle p_j h,p_j h\rangle=\langle h,p_j h\rangle=[h,p_j h]=[\delta b,p_j]=[b,\partial p_j h]. \end{array}$

If the support of ${h}$ does not meet ${\partial X_j}$, then ${\partial p_j h=p_j \partial h}$, and thus ${[b,\partial p_j h]=0}$ and ${p_j h=0}$. Thus ${K\cap ker(p_j)}$ contains all the kernels of linear froms “evaluation on simplices of ${\partial X_j}$”. This yield the codimension estimate.

Since ${codim(K\cap ker(p_j))=dim(p_j(K))}$, we see that

$\displaystyle \begin{array}{rcl} dim_{\Gamma}(K)\leq\frac{|\partial F_j|}{|F_j|} \end{array}$

which tends to 0, so ${dim_{\Gamma}(K)=0}$ and the Lemma is proved.

3. Euler-Poincaré characteristic

Theorem 2 (Atiyah)

$\displaystyle \begin{array}{rcl} \chi(\Gamma\setminus L)=\sum_i (-1)^i b_i(L,\Gamma)=\sum_i (-1)^i\beta_i(L,\Gamma). \end{array}$

I will prove a stronger inequality,

3.1. Morse inequalities

Theorem 3 For all ${n\leq dim(L)}$,

$\displaystyle \begin{array}{rcl} \sum_{i=0}^n (-1)^{n-i} \alpha_i\geq\sum_{i=0}^n (-1)^{n-i}\beta_i(L,\Gamma), \end{array}$

where ${\alpha_i}$ is the number of ${k}$-simplices in ${\Gamma\setminus L}$.

Proof: usual diagram chasing.

3.2. Consequences

Corollary 4 If ${\Gamma}$ is finitely generated and infinite, with ${g}$ generators, ${\beta_1(\Gamma)\leq g-1}$.

If ${\Gamma}$ is finitely presented with ${r}$ generators, then

$\displaystyle \begin{array}{rcl} \beta_2(\Gamma)-\beta_1(\Gamma)+\beta_0(\Gamma)\leq r-g+1. \end{array}$