Amenability and profinite completions of finitely generated groups

IF 0.6 3区数学 Q3 MATHEMATICS Groups Geometry and Dynamics Pub Date : 2021-06-16 DOI:10.4171/ggd/732

Steffen Kionke, E. Schesler

引用次数: 3

Abstract

This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability. We construct a finitely generated, residually finite, amenable group $A$ and an uncountable family of finitely generated, residually finite non-amenable groups all of which are profinitely isomorphic to $A$. All of these groups are branch groups. Moreover, picking up Grothendieck's problem, the group $A$ embeds in these groups such that the inclusion induces an isomorphism of profinite completions. In addition, we review the concept of uniform amenability, a strengthening of amenability introduced in the 70's, and we prove that uniform amenability indeed is detectable from the profinite completion.

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有限生成群的可修性与profinite完备

本文探讨了有限生成剩余有限群的有限商与可修性概念之间的相互作用。我们构造了一个有限生成、剩余有限、可服从群$a$和一个不可数族的有限生成、残余有限、不可服从群，所有这些群都与$a$同构。所有这些组都是分支组。此外，在Grothendieck的问题上，群$A$嵌入到这些群中，使得包含引起profinite完备的同构。此外，我们回顾了均匀可适性的概念，这是70年代引入的可适性增强，我们证明了从profinite完成中确实可以检测到均匀可适度。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Groups Geometry and Dynamics 数学-数学

CiteScore

1.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Groups, Geometry, and Dynamics is devoted to publication of research articles that focus on groups or group actions as well as articles in other areas of mathematics in which groups or group actions are used as a main tool. The journal covers all topics of modern group theory with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions, probabilistic and analytical methods, interaction with ergodic theory and operator algebras, and other related fields. Topics covered include: geometric group theory; asymptotic group theory; combinatorial group theory; probabilities on groups; computational aspects and complexity; harmonic and functional analysis on groups, free probability; ergodic theory of group actions; cohomology of groups and exotic cohomologies; groups and low-dimensional topology; group actions on trees, buildings, rooted trees.

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