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 70s, and we prove that uniform amenability is indeed detectable from the profinite completion.
Steffen Kionke; Eduard Schesler. Amenability and profinite completions of finitely generated groups. Groups, geometry, and dynamics, Tome 17 (2023) no. 4, pp. 1235-1258. doi: 10.4171/ggd/732
@article{10_4171_ggd_732,
author = {Steffen Kionke and Eduard Schesler},
title = {Amenability and profinite completions of finitely generated groups},
journal = {Groups, geometry, and dynamics},
pages = {1235--1258},
year = {2023},
volume = {17},
number = {4},
doi = {10.4171/ggd/732},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/732/}
}
TY - JOUR
AU - Steffen Kionke
AU - Eduard Schesler
TI - Amenability and profinite completions of finitely generated groups
JO - Groups, geometry, and dynamics
PY - 2023
SP - 1235
EP - 1258
VL - 17
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/732/
DO - 10.4171/ggd/732
ID - 10_4171_ggd_732
ER -
%0 Journal Article
%A Steffen Kionke
%A Eduard Schesler
%T Amenability and profinite completions of finitely generated groups
%J Groups, geometry, and dynamics
%D 2023
%P 1235-1258
%V 17
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/732/
%R 10.4171/ggd/732
%F 10_4171_ggd_732
