Soluble groups with no $\mathbb{Z}\wr \mathbb{Z}$ sections

Jacoboni, Lison; Kropholler, Peter

Annales Henri Lebesgue, Volume 3 (2020), pp. 981-998.

In this article, we examine how the structure of soluble groups of infinite torsion-free rank with no section isomorphic to the wreath product of two infinite cyclic groups can be analysed. As a corollary, we obtain that if a finitely generated soluble group has a defined Krull dimension and has no sections isomorphic to the wreath product of two infinite cyclic groups then it is a group of finite torsion-free rank. There are further corollaries including applications to return probabilities for random walks. The paper concludes with constructions of examples that can be compared with recent constructions of Brieussel and Zheng.

Dans cet article, nous établissons un théorème de structure pour les groupes résolubles de rang sans torsion infini et sans section isomorphe au produit en couronne de deux groupes cycliques infinis. Nous obtenons le corollaire suivant : si un groupe résoluble de type fini sans telle section admet une dimension de Krull alors il est de rang sans torsion fini. D’autres corollaires sont également déduits, en particulier une application aux probabilités de retour des marches aléatoires. L’article se termine avec la construction d’exemples qui peuvent être comparés avec des travaux récents de Brieussel et Zheng.

Classification: 20J05, 20E22
Keywords: wreath products, Krull dimension, soluble groups, torsion-free rank

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