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}.


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