Lück’s approximation theorem
For residually finite groups, -Betti numbers can be obtained as limits of ordinary Betti numbers.
Theorem 1 (Lück) Let be decreasing finte index normal subgroups of . Assume that
Assume that acts freely cocompactly on some simplicial complex . Then, for all ,
1. Preparation: counting roots
Let be a monic integral polynomial.
Indeed, the product of nonzero roots equals the first nonzero coefficient of , whose absolute value is .
Write , where is the orthogonal projector from cochains to harmonic cochains. The usual Betti number has a similar expression, except that is replaced with -invariant.
Since it involves only a fixed neighborhood of each simplex, the Laplacian does not distinguish from -invariant cochains. This holds as well for every polynomial in the Laplacian. Therefore, for large enough,
Let be an interval. Let denote the sum of the eigenspaces of relative to eigenvalues belonging to . The orthogonal projection onto is denoted by . Let
Let , and be the corresponding objects in . is increasing and
Our discussion of monic polynomials shows that
where is an upper bound on . This does not depend on , and is the number of orbits of -simplices in .
The idea is to approximate the indicator function of an interval by polynomials. Forevery , there exists a polynomial such that
The same inequality holds for operators and for traces,
This shows that tends to as tends to 0.
Thanks to the monic estimate, tends to 0 as tends to 0 uniformly in . Therefore converges uniformly to as tends to 0. This proves that tends to .
Theorem 2 (Bergeron-Gaboriau 2004) Let be decreasing finite index subgroups of . Then
where is a compact laminated space with free action.
where is the rooted tree associated to the sequence . Here, we use the extension of to equivalence relations, an elaboration on Connes’ work for foliations due to Gaboriau.
In case action on is free and is contractible, .
Connection with Lécureux’s talk. Each defines an (atomic) IRS. The limiting Betti number depends only on the limiting IRS, which is the IRS associated to the action on .