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.


\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}.


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