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

Jacoboni, Lison; Kropholler, Peter

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

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.

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.

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

[BCGS14] Bieri, Robert; Cornulier, Yves; Guyot, Luc; Strebel, Ralph Infinite presentability of groups and condensation, J. Inst. Math. Jussieu, Volume 13 (2014) no. 4, pp. 811-848 | DOI | MR | Zbl

[BGS17] Belyaev, Vasilĭ A.; Grigorchuk, Rostislav I.; Shumyatsky, Pavel On just-infiniteness of locally finite groups and their $C^{*}$-algebras, Bull. Math. Sci., Volume 7 (2017) no. 1, pp. 167-175 | DOI | MR

[Bie81] Bieri, Robert Homological dimension of discrete groups, Queen Mary College mathematics notes, Queen Mary College, Department of Pure Mathematics, 1981 | Zbl

[BMMN97] Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. Notes on infinite permutation groups, Texts and Readings in Mathematics, 12, Hindustan Book Agency, 1997 (co-published by Springer-Verlag, Berlin, Lecture Notes in Mathematics, vol. 1698) | MR | Zbl

[Bri15] Brieussel, Jérémie About the speed of random walks on solvable groups (2015) (https://arxiv.org/abs/1505.03294)

[BS78] Bieri, Robert; Strebel, Ralph Almost finitely presented soluble groups, Comment. Math. Helv., Volume 53 (1978) no. 2, pp. 258-278 | DOI | MR | Zbl

[BZ15] Brieussel, Jérémie; Zheng, Tianyi Speed of random walks, isoperimetry and compression of finitely generated groups (2015) (https://arxiv.org/abs/1510.08040) | Zbl

[Cor19] Cornulier, Yves Counting submodules of a module over a noetherian commutative ring, J. Algebra, Volume 534 (2019), pp. 392-426 | DOI | MR | Zbl

[Hal54] Hall, Philip Finiteness conditions for soluble groups, Proc. Lond. Math. Soc., Volume 4 (1954) no. 1, pp. 419-436 | DOI | MR | Zbl

[Jac19] Jacoboni, Lison Metabelian groups with large return probability, Ann. Inst. Fourier, Volume 69 (2019) no. 5, pp. 2121-2167 | DOI | MR | Zbl

[Kro84] Kropholler, Peter H. On finitely generated soluble groups with no large wreath product sections, Proc. Lond. Math. Soc., Volume 49 (1984) no. 1, pp. 155-169 | DOI | MR | Zbl

[Kro85] Kropholler, Peter H. A note on the cohomology of metabelian groups, Math. Proc. Camb. Philos. Soc., Volume 98 (1985) no. 3, pp. 437-445 | DOI | MR | Zbl

[LR04] Lennox, John C.; Robinson, Derek J. S. The theory of infinite soluble groups, Oxford Mathematical Monographs, Clarendon Press, 2004 | Zbl

[Mal51] Mal’tsev, Anatoliĭ I. On some classes of infinite soluble groups, Mat. Sb., N. Ser., Volume 28(70) (1951) no. 3, pp. 567-588

[MR87] McConnell, John C.; Robson, J. Chris Noncommutative Noetherian rings, Pure and Applied Mathematics, John Wiley & Sons, 1987 (With the cooperation of L. W. Small, A Wiley-Interscience Publication) | Zbl

[PSC03] Pittet, Christophe; Saloff-Coste, Laurent Random walks on finite rank solvable groups, J. Eur. Math. Soc. (JEMS), Volume 5 (2003) no. 4, pp. 313-342 | DOI | MR

[Rob96] Robinson, Derek J. S. A Course in the Theory of Groups, Graduate Texts in Mathematics, 80, Springer, 1996 | MR

[RW84] Robinson, Derek J. S.; Wilson, John S. Soluble groups with many polycyclic quotients, Proc. London Math. Soc., Volume 48 (1984) no. 2, pp. 193-229 | DOI | MR | Zbl

[Tes16] Tessera, Romain The large-scale geometry of locally compact solvable groups, Int. J. Algebra Comput., Volume 26 (2016) no. 2, pp. 249-281 | DOI | MR | Zbl

[Tus03] Tushev, Anatolii V. On deviation in groups, Ill. J. Math., Volume 47 (2003) no. 1-2, pp. 539-550 | DOI | MR | Zbl