The universal valued Abelian groups of Niemiec are Lévy, strongly exotic, extremely amenable, and 𝔾r(0) is generically monothetic
Abstract
We prove various properties of the universal valued Abelian groups 𝔾r(N) constructed by Niemiec, for r∈{1,∞} and N∈{0,2,3,…}. First we show that the completion of Γ0=⨁n∈ℕℚ/ℤ (for N=0) or ΓN=⨁n∈ℕℤ/Nℤ (for N≥ 2) with respect to a generic invariant metric, bounded by $1$ when r=1, is isometrically group-isomorphic to 𝔾r(N), recovering a result of Doucha in the case when $r=\infty $ and N=0. This confirms an expectation of Doucha for the group ΓN. We combine this genericity result with a criterion of Melleray and Tsankov concerning the extreme amenability of the generic completion of a countable group to obtain the extreme amenability of 𝔾r(N). We then establish the strictly stronger properties that these groups are Lévy and strongly exotic. We conclude by showing that 𝔾r(0) is also monothetic, from which we obtain the existence of a monothetic group structure on the Urysohn sphere, giving a bounded version of a result by Cameron and Vershik and answering a question of Niemiec.
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BibTeX
@misc{codenotti2026universal,
title = {{The universal valued Abelian groups of Niemiec are Lévy, strongly exotic, extremely amenable, and $\mathbbG_r(0)$ is generically monothetic}},
author = {Alessandro Codenotti and Martino Lupini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25151},
eprint = {2609.25151},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25151},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - The universal valued Abelian groups of Niemiec are Lévy, strongly exotic, extremely amenable, and 𝔾_r(0) is generically monothetic
AU - Alessandro Codenotti
AU - Martino Lupini
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25151
UR - https://arxiv.org/abs/2609.25151
AB - We prove various properties of the universal valued Abelian groups 𝔾_r(N) constructed by Niemiec, for r∈{1,∞} and N∈{0,2,3,…}. First we show that the completion of Γ_0=⨁_n∈ℕℚ/ℤ (for N=0) or Γ_N=⨁_n∈ℕℤ/Nℤ (for N≥ 2) with respect to a generic invariant metric, bounded by $1$ when r=1, is isometrically group-isomorphic to 𝔾_r(N), recovering a result of Doucha in the case when $r=\infty $ and N=0. This confirms an expectation of Doucha for the group Γ_N. We combine this genericity result with a criterion of Melleray and Tsankov concerning the extreme amenability of the generic completion of a countable group to obtain the extreme amenability of 𝔾_r(N). We then establish the strictly stronger properties that these groups are Lévy and strongly exotic. We conclude by showing that 𝔾_r(0) is also monothetic, from which we obtain the existence of a monothetic group structure on the Urysohn sphere, giving a bounded version of a result by Cameron and Vershik and answering a question of Niemiec.
ER -