The large-scale geometry of locally compact solvable groups

Tessera, Romain

doi:10.1142/s0218196716500120

Cited by 3 publications

(3 citation statements)

References 88 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Theorem B also has the following consequence for random walks on soluble linear groups. We refer for example to the survey of Tessera [Tes16] for background and definitions. In [Jac19], a group is said to have large return probability whenever its return probability is equivalent to exp(−n 1 3 ).…”

Section: The Main Structure Theoremmentioning

confidence: 99%

Soluble groups with no ℤ≀ℤ sections

Jacoboni¹,

Kropholler²

2020

Annales Henri Lebesgue

View full text Add to dashboard Cite

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. Résumé.-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.

show abstract

Section: The Main Structure Theoremmentioning

confidence: 99%

Soluble groups with no ℤ≀ℤ sections

Jacoboni¹,

Kropholler²

2020

Annales Henri Lebesgue

View full text Add to dashboard Cite

show abstract

“…Solvable Groups. For this next section, see [14,20,21,22] for a more thorough discussion about solvable groups.…”

Section: We Define Rfmentioning

confidence: 99%

Residual finiteness and strict distortion of cyclic subgroups of solvable groups

Pengitore¹

2020

Journal of Algebra

View full text Add to dashboard Cite

We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of solvable groups which include polycyclic groups with a nontrivial exponential radical and the metabelian Baumslag-Solitar groups, we improve the lower bounds found in the literature. Additionally, for the class of residually finite, finitely generated solvable groups of infinite Prüfer rank that satisfy the conditions of our theorem, we provide the first nontrivial lower bounds.

show abstract

“…The theorem also has the following consequence for random walks on soluble linear groups. We refer for example to the survey of Tessera [19] for background and definitions. In [10], a group is said to have large return probability whenever its return probability is equivalent to exp(−n 1 3 ).…”

Section: The Main Structure Theoremmentioning

confidence: 99%

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

Jacoboni¹,

Kropholler²

2018

Preprint

View full text Add to dashboard Cite

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. Groupes résolubles sans section isomorphe à Z ≀ Z RésuméDans cet article, nous donnons 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. En conséquence, 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.

show abstract

The large-scale geometry of locally compact solvable groups

Cited by 3 publications

References 88 publications

Soluble groups with no ℤ≀ℤ sections

Soluble groups with no ℤ≀ℤ sections

Residual finiteness and strict distortion of cyclic subgroups of solvable groups

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

Contact Info

Product

Resources

About