** A random walk around soluble groups theory **

Notes by Gabriel Pallier.

**1. Minimax groups **

Recall that for any prime number , the Pruefer group may be defined as . Pruefer group are also called quasi-cyclic, as any proper subgroup of a Pruefer group is finite and cyclic.

**Definition 1** * A polyminimax group is one with a composition series *

*
** where 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 is isomorphic to a direct product . As such, it is an abelian, non finitely-generated polyminimax group.

**Theorem 2 (Kropholler 1984)** * If is a finitely generated soluble group, then *

*
*
- {Either, is polyminimax, or}
- { has a lamplighter section for some prime .}

* *

Classical known results : if is polycyclic, then

- {Hall 1960 : is Noetherian.}
- {Hall-Roseblade 1973 : irreducible -modules are finite.}

**Theorem 3 (Jacoboni, 2016)** * Let be a finitely generated metabelian group. Assume that has Krull dimension greater or equal to . Then has a section isomorphic to or , where denotes the free group on two generators. *

The assumption on the Krull dimension of may be expressed as follows : any sequence

with abelian and makes a Noetherian module over the finitely generated commutative ring (indeed acts on via the conjugation action of on ). Then it is sufficient that have Krull dimension at leat .

For polyminimax groups, the return probability (or if one authorizes to pause at some times) is bounded below by

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 ( metabelian, finitely generated with Krull dimension ), a reverse inequality

**2. Finitely Generated groups with no sections **

Let have finite Hirsch length, i.e.

(Observe that only the sections increment the Hirsch length).

**Theorem 4 (Kropholler-Jacoboni, 2016)** * If is soluble, finitely generated, without section and , then admits a quotient as follows:*

*
* with , non polyminimax, a torsion-free abelian, , decomposing as

* , . *

**Corollary 5** * If is finitely generated, with a well-defined Krull dimension, then either has a section, or . *

**3. Open problem **

Let be polyminimax and a simple -module. Is elementary abelian ? Here Hall’s method would work only provided that is Noetherian. A necessary condition for this is that be amenable, since Bartholdi (2016) produces injection for non amenable .

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## About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
http://www.math.ens.fr/metric2011/