## Gabriel Pallier’s notes of Peter Kropholler’s Cambridge lecture 12-05-2017

A random walk around soluble groups theory

Notes by Gabriel Pallier.

1. Minimax groups

Recall that for any prime number ${p}$, the Pruefer group ${C_{p^\infty}}$ may be defined as ${\mathbb{Z}[1/p] / \mathbb{Z}}$. Pruefer group are also called quasi-cyclic, as any proper subgroup of a Pruefer group is finite and cyclic.

Definition 1 A polyminimax group ${G}$ is one with a composition series

$\displaystyle 1 = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G,$

where ${G_{i+1}/G_i}$ is either cyclic or quasicyclic.

This defines a class of groups, intermediate between polycyclic and soluble.

Example to be thought of : the additive group of the ring ${\mathbb{Z} \left[ \frac{1}{4+7i} \right]}$ is isomorphic to a direct product ${C_{5^{\infty}} \times C_{13^{\infty}}}$. As such, it is an abelian, non finitely-generated polyminimax group.

Theorem 2 (Kropholler 1984) If ${G}$ is a finitely generated soluble group, then

• {Either, ${G}$ is polyminimax, or}
• {${G}$ has a lamplighter section ${C_p \, \mathcal{o} \, \mathbb{Z}}$ for some prime ${p}$.}

Classical known results : if ${G}$ is polycyclic, then

1. {Hall 1960 : ${\mathbb{Z} G}$ is Noetherian.}
2. {Hall-Roseblade 1973 : irreducible ${\mathbb{Z} G}$-modules are finite.}

Theorem 3 (Jacoboni, 2016) Let ${G}$ be a finitely generated metabelian group. Assume that ${G}$ has Krull dimension greater or equal to ${2}$. Then ${G}$ has a section isomorphic to ${\mathbb{Z} \, \mathcal{o} \, \mathbb{Z}}$ or ${F / (F'^p) F''}$, where ${F}$ denotes the free group on two generators.

The assumption on the Krull dimension of ${G}$ may be expressed as follows : any sequence

$\displaystyle A \rightarrow G \rightarrow Q\rightarrow 1,$

with abelian ${A}$ and ${Q}$ makes ${A}$ a Noetherian module over the finitely generated commutative ring ${\mathbb{Z} Q}$ (indeed ${Q}$ acts on ${A}$ via the conjugation action of ${G}$ on ${A}$). Then it is sufficient that ${A}$ have Krull dimension at leat ${2}$.

For polyminimax groups, the return probability ${p(2n)}$ (or ${p(n)}$ if one authorizes to pause at some times) is bounded below by

$\displaystyle p(n) \geq e^{-n^{1/3}}.$

This was claimed by Pittet and Saloff-Coste in 2004, however with a mistake in the proof. Jacoboni proves under the same assumption as in theorem 3 (${G}$ metabelian, finitely generated with Krull dimension ${\geq 2}$), a reverse inequality

$\displaystyle p(n) \leq \exp ( -n^{1/3} (\log n)^{2/3}).$

2. Finitely Generated groups with no ${\mathbb{Z} \, \mathcal{o} \, \mathbb{Z}}$ sections

Let ${G}$ have finite Hirsch length, i.e.

$\displaystyle h(G) := \sum_{i \geq 0} \dim_{\mathbb{Q}} \left( G^{(i)} / G^{(i+1)} \otimes \mathbb{Q} \right) < +\infty.$

(Observe that only the ${\mathbb{Z}}$ sections increment the Hirsch length).

Theorem 4 (Kropholler-Jacoboni, 2016) If ${G}$ is soluble, finitely generated, without ${\mathbb{Z} \, \mathcal{o} \, \mathbb{Z}}$ section and ${h(G) = + \infty}$, then ${G}$ admits a quotient ${\overline{G}}$ as follows:

$\displaystyle A \rightarrow \overline{G} \rightarrow Q\rightarrow 1,$

with ${h(Q) < + \infty}$, ${Q}$ non polyminimax, ${A}$ a torsion-free abelian, ${h(A) = + \infty}$, ${A}$ decomposing as

$\displaystyle A = \bigoplus_{n \in \mathbb{Z}} A_n,$

${A_n \triangleleft \overline{G}}$, ${h(A_n) < + \infty}$.

Corollary 5 If ${G}$ is finitely generated, with a well-defined Krull dimension, then either ${G}$ has a ${\mathbb{Z} \, \mathcal{o} \, \mathbb{Z}}$ section, or ${h(G) < + \infty}$.

3. Open problem

Let ${G}$ be polyminimax and ${M}$ a simple ${\mathbb{Z} G}$-module. Is ${M}$ elementary abelian ? Here Hall’s method would work only provided that ${\mathbb{Z} G}$ is Noetherian. A necessary condition for this is that ${G}$ be amenable, since Bartholdi (2016) produces injection ${\mathbb{Z} G^n \hookrightarrow \mathbb{Z} G^{n-1}}$ for non amenable ${G}$.