Growth of torsion homology in amenable groups (Nikolay Nikolov)

Let \( G \) be a discrete group and \( \widetilde X \) a simply-connected CW-complex with a free \( G \)-action, and \( X \) the quotient \( G \backslash \widetilde X \). A particular case is when \( \widetilde X \) is a classifying space for \( G \), i.e. contarctible.

The aim is to study the homology groups \( H_n(\cdot; {\mathbb Z}) \) for finite covers of \( X \). For this suppose that the \( n + 1 \)-skeleton of \( X \) is finite, and take a residual chain \( G_0 = G \supset G_1 \supset \cdots \) of normal, finite-index subgroups of \( G \) such that \( \bigcap_i G_i = \{ 1 \} \). Denote \( X_i = G_i \backslash \widetilde X \). The Lûck Approximation Theorem states that:
\[
\lim_{i \to +\infty} \frac{\mathrm{rank}_{\mathbb Z} H_n(X_i ; {\mathbb Z})} {|G / G_i|} = b_n^{(2)}(\widetilde X \to X).
\]
The question motivating the rest of the talk will be to estimate the growth of \( t(H_n(X_i ; {\mathbb Z})) \) (where \( t(A) \) is the size of the torsion subgroup of a finitely generated Abelian group \( A \)). In full generality it is possible to say that \( \log(t(H_n(X_i ; {\mathbb Z})) / |G/G_i| \) is bounded.

Theorem (Kar–Kropholler–Nikolov): Suppose that \( G \) is amenable and that \( H_n(\widetilde X; {\mathbb Z}) = 0 \) (for example \( \widetilde X \) is contractible). Then
\[
\lim_{i \to +\infty} \frac{t(H_n(X_i; {\mathbb Z}))}{|G/G_i|} = 0.
\]

Lück has proven that the same conclusion holds in the case where \( G \) contains an infinite elementary amenable normal subgroup. A natural question is then:

Does the same conclusion holds if \( G \) is only supposed to contain an inifinite amenable normal subgroup?

This was proven by Sauer in the case where \( \widetilde X \) is a simply-connected manifold with a proper, free, cocompact \( G \)-action.

Proof of the theorem

The main idea (already used in previous work by Abért–Jaikin-Zapirain–Nikolov), is to use the following result.

Theorem (Benjamin Weiss): Let \( G \) be a finitely generated amenable group with a residual chain \( G_i \). Then there exists a sequence \( F_i \) such that

  • \( G = \bigsqcup_{g \in G_i} gF_i \);
  • \( (F_i ) \) is a Følner sequence.

Let \( S \) be any finite subset of \( G \). Weiss’ theorem yields for each \( i \) a finite subcomplex \( \widetilde D_i \subset \widetilde X \) which contains a subcomplex \( \widetilde J_i \) such that \( \partial_S \widetilde J_i \subset \widetilde D_i \) and \( |J_i| / |\widetilde D_i| \underset{i \to +\infty}{\rightarrow} 1 \).

Let \( p_i : \widetilde X \to X_i \) be the covering map and \( J_i = p_i(\widetilde J_i) \). Then for large \( i \) the following exact sequence holds:
\[
H_n(\widetilde X; {\mathbb Z})|_{\widetilde J_i} \overset{p_i}{\to} H_n(X_i; {\mathbb Z}) \to H_n(X_i, J_i; {\mathbb Z})
\]
where \( H_n(\widetilde X; {\mathbb Z})|_{\widetilde J_i} \) is the submodule genertated by those classes in \( H_n(\widetilde X; {\mathbb Z}) \) that can be represented by chains with support in \( \widetilde J_i \). Indeed, taking \( S \) to be the union of supports of all coefficients of a matrix representing \( \tilde d_{n+1} \), we see that for \( i \) large enough so that \( \partial_S \widetilde J_i \subset \widetilde D_i \) we have \( d_{n+1} \widetilde J_i^{n+1} \subset \widetilde D_i^n \), so that if we have a chain \( c \in C_n(X_i; {\mathbb Z}) \) with support in \( J_i \), by lifting it to \( \widetilde D_i \) we retain a chain.

Now if \( H_n(\widetilde X; {\mathbb Z}) = 0 \) we get that
\[
H_n(X_i; {\mathbb Z}) \hookrightarrow H_n(X_i, J_i; {\mathbb Z}).
\]
Since \( H_n(X_i, J_i; {\mathbb Z}) \hookrightarrow C_n( D_i \setminus J_i ) \) the torsion on the left-hand side ic \( o( \|d_{n+1}|^{|D_i \setminus J_i|} ) \) and thus
\[
\frac{\log t(H_n(X_i; {\mathbb Z}))} {|G/G_i|} \ll \frac{|D_i \setminus J_i|}{|G/G_i|} \underset{i\to+\infty}{\rightarrow} 0.
\]

Weakening the hypotheses

Is the hypothesis that \( X \) has a finite \( n+1 \)-skeleton needed to rech the conclusion of the theorem? In general yes, but in some cases one can prove subexponential growth of torsion without it.

Theorem (Kar–Kropholler–Nikolov): Let \( G \) be a metabelian group and let \( G_i \subset G \) be finite-index subgroups satisfying the condition that:
\[
| [G, G] / (G_i \cap [G, G]) | \to +\infty.
\]
Then
\[
\lim_{i \to +\infty} \frac{t(H_n(X_i; {\mathbb Z}))}{|G/G_i|} = 0.
\]

Theorem: For any function \( f: \mathbb N \to \mathbb N \) there exists a finitely generated solvable group \( G \) with a residual chain \( G_i \) such that \( t(H_1(G_i; {\mathbb Z})) \ge f(i) \).