## Notes of Damien Gaboriau’s lecture nr 1

${\ell^2}$ Betti numbers

Started with M. Atiyah (1976) for manifolds. Very soon (1979), Connes obtained a version for foliations. In 1986, Cheeger and Gromov extended the notion to arbitrary countable groups.

1. Von Neumann dimension

Murray-von Neumann 1944. Let ${V}$ be a closed ${\Gamma}$-invariant subspace in ${\bigoplus_{i=1}^k \ell^2(\Gamma)}$. A real number ${dim_\Gamma(V)}$ can be defined in such of way that

1. ${dim_\Gamma(V)=0\Leftrightarrow V=0}$.
2. ${dim_\Gamma(\ell^2(\Gamma))=1}$.
3. If ${f:V\rightarrow W}$ is bounded and equivariant, ${dim_\Gamma(V)=dim_\Gamma(ker(f))+dim_\Gamma(im(f))}$.

1.1. ${\Gamma}$-trace

Let ${a}$ be an operator which commutes with the left regular representation. Express it in the canonical basis ${\{1_g\,;\,g\in\Gamma\}}$. The diagonal elements ${\langle a1_g,1_g\rangle}$ are all equal. Denote this value by ${\tau(a)}$. Check that ${\tau(ab)=\tau(ba)}$.

1.2. Case ${V\subset\ell^2(\Gamma)}$

Orthogonal projection onto ${V}$ is an element ${p}$ which commutes with the left regular representation. So define ${dim_\Gamma(V)=\tau(p)}$.

Since ${p}$ is an orthogonal projector,

$\displaystyle \langle p1_e,1_e\rangle=\langle p^2 1_e,1_e\rangle=\langle p1_e,p1_e\rangle=|p1_e|^2.$

Therefore ${dim_\Gamma(V)=0}$ implies ${p1_e=0}$, and thus ${p=0}$, ${V=0}$.

1.3. Exercise

If ${\Gamma={\mathbb Z}}$, Fourier transform maps ${\ell^2({\mathbb Z})}$ to ${L^2(S^1)}$, the ${{\mathbb Z}}$ action is by multiplication with function ${z^k}$. ${{\mathbb Z}}$-equivariant operators are multiplication operators with bounded functions ${M_h}$. ${1_e}$ corresponds to constant function 1. ${\langle M_h 1_e,1_e\rangle=\int_{S^1}h(x)\,dx}$. Orthogonal projections are indicator functions of Borel sets ${B}$, invariant subspaces are of the form ${L^2(B)\subset L^2(S^1)}$, ${\tau(1_B)=}$ measure of ${B}$.

1.4. General case

View projector on ${\bigoplus_{i=1}^k \ell^2(\Gamma)}$ as a matrix in block form. Diagonal blocks turn out to be ${\Gamma}$-equivariant. Define ${Trace (p)=\sum_{i=1}^k \tau(p_{ii})}$.

1.5. Homology

Let ${L}$ be a countable simplicial complex with free cocompact ${\Gamma}$-action. ${\ell^2}$ chains make sense, therefore ${\ell^2}$-homology is defined. Reduced ${\ell^2}$-homology is isomorphic to the subspace of ${\ell^2}$ harmonic ${n}$-chains ${\mathcal{H}^{(2)}_n}$. This can be viewed as a ${\Gamma}$-invariant subspace in finitely many copies of ${\ell^2(\Gamma)}$. Define

$\displaystyle \begin{array}{rcl} \beta_n(L,\Gamma)=dim_\Gamma(\mathcal{H}^{(2)}_n). \end{array}$

Observe that

$\displaystyle \begin{array}{rcl} \beta_n(L,\Gamma)=dim_\Gamma(ker(\partial_n))-dim_\Gamma(im((\partial_{n-1})). \end{array}$

Proposition 1 For all ${p}$-connected ${L}$, the ${\ell^2}$-Betti numbers are the same up to ${p}$. This defines ${\beta_n(\Gamma)}$.

1.6. Cheeger-Gromov’s definition

For a general group ${\Gamma}$, an infinite dimensional contractible simplicial complex ${L}$ (with free action of ${\Gamma}$) may be needed. Exhaust it with co-finite ones, and take a limit of ${dim_\Gamma(im (H_n(L_j)\rightarrow H_n(L_{j+1})))}$. Does not depend on exhaustion.

2. Some values of ${\ell^2}$ Betti numbers

Finite groups: all 0 but the first, ${\beta_0=\frac{1}{|\Gamma|}}$.

Infinite groups: ${\beta_0=0}$.

Free group ${\mathbf{F}_n}$: ${0,n-1,0,\ldots}$.

${\Gamma}$ generated by ${g}$ elements: ${\beta_1(\Gamma)\leq g}$.

${\Gamma}$ infinite amenable: ${0,0,\ldots}$ (I will prove this).

Surface group of genus ${g}$: ${0,2g-2,0,0,\ldots}$.

Lattices in ${U(p,q)}$, ${Sp(p,q)}$,…: exactly one nonzero ${\ell^2}$ Betti number.

2.1. Some results

Passing to finite index subgroup multiplies ${\ell^2}$ Betti numbers by index.

Euler characteristic ${\chi(\Gamma)=\sum(-1)^n\beta_n(\Gamma)}$.

Lück’s approximation theorem: see lecture nr 3.

2.2. Atiyah’s conjecture

Atiyah: if torsion free, are ${\ell^2}$ Betti numbers integers ?

The generalized form (under restrictions on torsion) has been disproved, at least for actions. Grigorchuk-Zuk calculation for the lamplighter group.