Notes of Damien Gaboriau’s Lille lecture nr 3

Lück’s approximation theorem

For residually finite groups, ${\ell^2}$-Betti numbers can be obtained as limits of ordinary Betti numbers.

Theorem 1 (Lück) Let ${\Gamma_n}$ be decreasing finte index normal subgroups of ${\Gamma}$. Assume that

$\displaystyle \begin{array}{rcl} \bigcap_n \Gamma_n=\{e\}. \end{array}$

Assume that ${\Gamma}$ acts freely cocompactly on some simplicial complex ${L}$. Then, for all ${d}$,

$\displaystyle \begin{array}{rcl} \lim_{n\rightarrow\infty}\frac{1}{[\Gamma:\Gamma_n]}Betti_d(\Gamma_n\setminus L)=\beta_d(L,\Gamma). \end{array}$

1. Preparation: counting roots

Let ${M(T)=\prod(T-\lambda_i)}$ be a monic integral polynomial.

$\displaystyle \begin{array}{rcl} |\{\lambda_i\,;\,|\lambda_i|\in)0,\lambda)\}|\leq degree(M)\frac{\log D}{-\log\lambda}, \end{array}$

where ${D=\max|\lambda_i|}$.

Indeed, the product of nonzero roots equals the first nonzero coefficient of ${M}$, whose absolute value is ${\geq 1}$.

2. Proof

Write ${\beta_i(L,\Gamma)=\sum_{\sigma\subset D}\langle p_{ker}\sigma,\sigma\rangle}$, where ${p_{ker}}$ is the orthogonal projector from ${\ell^2}$ cochains to ${\ell^2}$ harmonic cochains. The usual Betti number has a similar expression, except that ${\ell^2}$ is replaced with ${\Gamma_n}$-invariant.

Since it involves only a fixed neighborhood of each simplex, the Laplacian does not distinguish ${\ell^2}$ from ${\Gamma_n}$-invariant cochains. This holds as well for every polynomial ${Q(\Delta)}$ in the Laplacian. Therefore, for ${n}$ large enough,

$\displaystyle \begin{array}{rcl} \frac{1}{[\Gamma:\Gamma_n]}Trace(Q(\Delta_n))=Trace_{\Gamma_n\setminus\Gamma}(Q(\Delta_n))=Trace_\Gamma(\Delta). \end{array}$

Let ${I\subset{\mathbb R}}$ be an interval. Let ${E_n(I)}$ denote the sum of the eigenspaces of ${\Delta_n}$ relative to eigenvalues belonging to ${I}$. The orthogonal projection onto ${E_n(I)}$ is denoted by ${\Pi_n(I)}$. Let

$\displaystyle \begin{array}{rcl} f_n(I)=\frac{1}{[\Gamma:\Gamma_n]}dim(E_n(I)). \end{array}$

Let ${E(I)}$, ${\Pi(I)}$ and ${f(I)}$ be the corresponding objects in ${\ell^2}$. ${f}$ is increasing and

$\displaystyle \begin{array}{rcl} \lim_{\lambda\rightarrow 0}f((0,\lambda))=0. \end{array}$

Our discussion of monic polynomials shows that

$\displaystyle \begin{array}{rcl} f_n((0,\lambda))\leq \alpha_d \frac{\log D}{-\log\lambda} \end{array}$

where ${D}$ is an upper bound on ${\|\Delta_n\|}$. This does not depend on ${n}$, and ${\alpha_d}$ is the number of orbits of ${d}$-simplices in ${L}$.

The idea is to approximate the indicator function of an interval by polynomials. Forevery ${\eta>0}$, there exists a polynomial ${Q_\eta}$ such that

$\displaystyle \begin{array}{rcl} 1_{[0,\eta)}\leq Q_\eta\leq (1+\eta)1_{[0,2\eta)}+\eta 1_{[0,T]}. \end{array}$

The same inequality holds for operators and for traces,

$\displaystyle \begin{array}{rcl} f([0,\eta))\leq Trace_\Gamma Q_\eta(\Delta)\leq(1+\eta)f([0,2\eta))+\eta \alpha_d. \end{array}$

This shows that ${Trace_\Gamma Q_\eta(\Delta)}$ tends to ${f(0)=\beta_d}$ as ${\eta}$ tends to 0.

Similarly,

$\displaystyle \begin{array}{rcl} f_n([0,\eta))\leq Trace_{\Gamma_n\setminus\Gamma} Q_\eta(\Delta_n)\leq(1+\eta)f_n([0,2\eta))+\eta \alpha_d. \end{array}$

Thanks to the monic estimate, ${f_n((0,\eta))}$ tends to 0 as ${\eta}$ tends to 0 uniformly in ${n}$. Therefore ${Trace_{\Gamma_n\setminus\Gamma} Q_\eta(\Delta_n)}$ converges uniformly to ${Betti_d(\Gamma_n\setminus L)}$ as ${\eta}$ tends to 0. This proves that ${Betti_d(\Gamma_n\setminus L)}$ tends to ${\beta_d}$.

3. Generalization

Theorem 2 (Bergeron-Gaboriau 2004) Let ${\Gamma_n}$ be decreasing finite index subgroups of ${\Gamma}$. Then

$\displaystyle \begin{array}{rcl} \lim_{n\rightarrow\infty}\frac{1}{[\Gamma:\Gamma_n]}Betti_d(\Gamma_n\setminus L)=\beta_d(\mathcal{L},\Gamma), \end{array}$

where ${\mathcal{L}}$ is a compact laminated space with free ${\Gamma}$ action.

$\displaystyle \mathcal{L}=\Gamma\setminus(\partial T)\times L,$

where ${T}$ is the rooted tree associated to the sequence ${\Gamma_n}$. Here, we use the extension of ${\beta_d}$ to equivalence relations, an elaboration on Connes’ work for foliations due to Gaboriau.

In case ${\Gamma}$ action on ${\partial L}$ is free and ${L}$ is contractible, ${\beta_d(\mathcal{L},\Gamma)=\beta_d(\Gamma)}$.

Connection with Lécureux’s talk. Each ${\Gamma_n}$ defines an (atomic) IRS. The limiting ${\ell^2}$ Betti number depends only on the limiting IRS, which is the IRS associated to the ${\Gamma}$ action on ${\partial T}$.