Lück approximation theorem

Let \( K \) be a finite CW-complex with residually finite fundamental group \( \Gamma = \pi_1(K) \). Let \( \Gamma=\Gamma_0 \supset \Gamma_1 \supset \cdots ) \) be a residual chain in \( \Gamma \), meaning that the \( \Gamma_n \) are finite-index, normal subgroups and \( \bigcap_n \Gamma_n = \{ 1 \} \). Denote by \( widetilde K \to K \) the universal cover and \( K_n = \Gamma_n \backslash \widetilde K \). Then the Lück approximation theorem states that the \( L^2 \)-Betti numbers of the covering \( \widetilde K \to K \) are given by :
\[
\beta_q^{(2)}(\widetilde K \to K) = \lim_{n\to+\infty} \frac {b_n(K_n; {\mathbb Q})} {|\Gamma / \Gamma_n|}.
\]
The following question is then very natural, and was apparently first asked by Farber around 1998:

Question: Can one prove a statement similar to Lück approximation for Betti numbers with coefficients in a field of positive characteristic?

A major step towards answering this question is to define an analogue for \( L^2 \)-Betti numbers for fields with positive characteristic, which has so far been done only in specific cases. An example is the following result.

Theorem (Elek–Szabó): Let \( k \) be a field. If \( \Gamma \) is a discrete, residually finite amenable group. Then for any finite CW-complex \( K \) with \( \pi_1(K) \cong \Gamma \) and any residual chain \( (\Gamma_n) \) the limit
\[
\lim_{n\to+\infty} \frac {b_n(K_n; k)} {|\Gamma / \Gamma_n|}
\]
exists. Moreover it is independent of the choice of \( (\Gamma_n) \).

Other results in this direction are due to Lackenby, Abért–Nikolov, Linnell–Lück–Sauer,…

Classical Atiyah problem

The classical Atiyah problem can be formulated as follows.

Atiyah problem: Fix a discrete group \( \Gamma \). What are the possible values for the \( \ell^2 \)-Betti numbers \( \beta^{(2)}(\widetilde K \to K) \) of \( \Gamma \)-covers? In particular, are they all rational?

In full generality the answer to the second question is very strongly « no ». The work of various people on this can be summerised in the following statement.

Theorem (Austin, Grabowski, Pichot–Schick–Zuk):

1. For any positive real number \( \alpha \) there exists a group \( G \) and a \( G \)-covering \( \widehat K \to K \) such that \( \beta_3^{(2)}(\widehat K \to K) = \alpha \).
2. If moreover \( \alpha \) is computable then we can take \( \widehat K \) to be simply connected in the previous point.

This statement is in fact really about group ring elements. The proof proceeds by showing that:

1. For any positive real number \( \alpha \) there exists a group \( G \) and \( M \in M_{n \times m}({\mathbb C} G) \) such that \( \dim_{\mathcal NG} \ker(M) = \alpha \).
2. If moreover \( \alpha \) is computable then we can take \( G \) to be finitely presented.

Homology gradients over finite fields

From now on \( k \) will be a finite field of characteristic at least 3 (it being of characteristic 2 causes some technical issues, but most of what follows goes through in general). Let \( G \) be an amenable group and \( T _in kG \). For \( G \) residually finite we can define :
\[
\dim_{kG} \ker(T) = \lim_{n\to+\infty} \frac{\dim_k(\ker(T_n)} {|\Gamma / \Gamma_n|}
\]
where \( (\Gamma_n) \) is a residual chain and \( T_n \in k(\Gamma/\Gamma_n) \) is the reduction of \( T \) (in general it is possible to define the right-hand side by using Følner sequences). For these homology gradients there is a result similar to the one in characteristic 0.

Theorem (Grabowski–Schick):

1. For any positive real number \( \alpha \) there exists an amenable group \( G \) and \( T \in kG) \) such that \( \dim_{kG} \ker(T) = \alpha \).
2. There exists a finitely presented amenable \( G \) and \( T \in kG \) with \( \dim_{kG} \ker(T) \in {\mathbb R} \setminus {\mathbb Q} \).

Later on it will be explained with point 2. above is much weaker than the corresponding statement in characteristic 0.

The same kind of techniques as used in the proof of the previous statement can also be used to give a counter example to a conjecture by A. Thom:

Theorem (Grabowski–Schick): Let \( G = {\mathbb Z}/2 \wr {\mathbb Z} \). There exists a matrix \( T \) with coefficients in the group ring \( {\mathbb Z} G \) such that the set \( \{ \dim_{\mathbb F_pG} \ker(T) \) is infinite (it does not « stabilise » to the von Neumann dimension \( \dim_{\mathcal NG} \ker(T) \)).

Note that for classical Betti numbers (i.e. for a matrix \( A \in M_{n\times m}({\mathbb Z}) \) it is clear that \( \dim_{\mathbb F_p} \ker A = \dim_{\mathbb C} \ker(A) \) for large enough \( p \). On the other hand the limit:
\[
\lim_{p \to +\infty} \dim_{\mathbb F_pG} \ker(T) = \dim_{\mathcal NG} \ker(T)
\]
is provable (?).

In view of this it is perhaps surprising that the following result holds.

Theorem (Grabowski–Schick): For \( G = {\mathbb Z}/2 \wr {\mathbb Z} \), any \( n, m \ge 1, T \in M_{n \times m}({\mathbb Q} G) \), the von Neumann dimension \( \dim_{\mathcal NG} \ker(T) \in {\mathbb Q} \).

On the other hand one can find such \( T \) with their \( \mathbb F_p \)-gradients not rational (Grabowski).

One of the main tools used in the proofs is the following interpretation of the von Neumann dimension of certain particular modules.

Theorem (Lehner–Neuhauser–Woess): Let \( \Gamma = \langle \gamma_1, \ldots, \gamma_r \rangle \) be a finitely generated group. For \( n \ge 1 \) let \( G_n = ({\mathbb Z}_2)^n \wr \Gamma \) and define \( A, T \in {\mathbb Q} G_n \) by:
\[
A = \frac 1 {2^n} \sum_{a \in ({\mathbb Z}/2)^n} a, \quad T = \sum_{i=1}^r (\gamma_i A + A\gamma_i^{-1}).
\]
The for large enough \( n \) it holds that:
\[
\dim_{\mathcal NG_n} \ker(T) = \sum_{\mathcal G} \frac {\mathrm P\mathcal G}{|\mathcal G} \dim_{\mathbb C} \ker(M_{\mathcal G})
\]

In this statement:

• The sum is over all finite graphs \( \mathcal G \);
• \( \mathrm P\mathcal G \) is the probability that the identity component of the identity in teh Cayley graph ofg \( \Gamma \) after percolation of parameter \( 2^{-n} \) is isomorphic to \( \mathcal G \);
• \( M_{\mathcal G} \) is the Markov operator associated to the simple random walk on \( \mathcal G \).

If \( \Gamma \) is amenable then the same formula holds with \( \mathcal NG_n, \mathcal {\mathbb C} \) replaced with \( \mathbb F_pG_n, \mathbb F_p \). It would be possible to obtain an optimal statement in 2. in positive characteristic if it was known to hold for nonamenable groups as well.