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.


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