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}

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/
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