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Erratum: metric geometry of locally compact groups by Yves Cornulier and Pierre de la Harpe
(Prepublished version: pdf version of July 2016,
233 pages; EMS Tracts in Mathematics Vol. 25, European Math.
Society, September 2016. Book at EMS'
site).
This page will be regularly updated. The authors are grateful to
people reporting any possible mistake or typo. If so please refer if
possible to the statement numbers (e.g. "3 lines before Proposition
3A6") since the page numbering differs between versions.
[04-2020] Prop. 3A6: it should be assumed that either g or g' is coarsely
Lipschitz.
[04-2020] Lemma 3B6(2) dc(x,y) should be dX(x,y)
[04-2020] Remark 3B8(1) ≤ cΦ should be ≤ Φ
[04-2020] Proposition 3C3. Not a mistake, but should rather be ≤c than ≤2c
[04-2020] Example 3C4(4)§2 "product a primes" should be "product of primes"
[04-2020] Example 3D6(2). The second assertion of this (2) is false: if
X is the set of square integers and Y the set of all
non-negative integers, mapping n to its square root is a
bijective 1-Lipschitz map, but the growth of X is roughly
√n while the growth of Y is roughly n. So in
the second assertion of (2), the inequality should be changed to
c''d(x,x')≤ d(f(x),f(x'))≤ cd(x,x'). In the proof of (2),
the changes are: in the displayed formula after "Suppose moreover...",
in indexing the last union, d(x,x')≤c(r+c'+s) should be
d(x,x')≤(r+c'+s)/c''. In the next sentence "Taking
cardinals...", cr+cc'+cs should be (r+c'+s)/c''.
[04-2020] Example 5.B.8. It is claimed that the usual Polish topology is the
unique non-discrete Hausdorff group topology on the infinite symmetric
group Sym(N), with reference to [KeRo] (Kechris-Rosendal). The
correct statement, which is written in [KeRo] is that it is the unique *separable*
Hausdorff group topology on Sym(N). Indeed, there exists a
non-discrete (and non-separable) Hausdorff group topology on Sym(N),
which is the topology induced by inclusion in Sym(2N) (this
observation is made in [BGP-12,§7]). That this is a Hausdorff group
topology is trivial, and that it is not discrete is a simple verification; it does not coincide with the usual topology because one can check elementarily that the finitary subgroup is closed.
[BGP-12]: T. Banakh, I. Guran, I. Protasov. Algebraically
determined topologies on permutation groups.
Topol. Appl. 159 (2012) 2258-2268.
[09-2023] (Addendum to the previous
item, 1) It can be checked that the number of Hausdorff group topologies
on Sym(N) is the maximum possible, namely 22c
[09-2023] (Addendum to the previous
item, 2) The original Polish topology on Sym(N) was originally
introduced by Luigi Onofri in: Teoria delle sostituzioni che operano su una
infinità numerable di elementi. Annali di Mat. (4) 4, 73-106 (1927) (and then forgotten and rediscovered in the 1950s).
[10-2024] Proposition 3.A.22(2):
"large-scale expansive" should be "coarsely expansive". (Thanks to Robert Tang for pointing this out.)