Amenability and profinite completions of finitely generated groups

2021/06/16 by Kionke, Steffen, Schesler, Eduard · 4 citations
#20E08 #20E18 #20E26 #43A07 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2106.08742

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.

Cited by

• Uniform amenability at infinity
• Infinitely presented simple groups separated by homological finiteness properties
• A branch group with unsolvable conjugacy problem
• Stabilized automorphism groups and full groups of odometers

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