\documentclass[JEP,XML,SOM,Unicode,published]{cedram}
\datereceived{2023-09-21}
\dateaccepted{2024-09-03}
\dateepreuves{2024-09-30}

\TralicsDefs{\addattributestodocument{type}{corrigendum}}
\Relation{JEP_2020__7__431_0}

\DeclareMathOperator{\Homeo}{Homeo}
\DeclareMathOperator{\Ends}{Ends}
\DeclareMathOperator{\Br}{Br}
\DeclareMathOperator{\CLO}{CCLO}
\DeclareMathOperator{\Sa}{Sa}

\theoremstyle{plain}
\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}

\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}

\datepublished{2024-10-08}
\begin{document}
\frontmatter
\title{Erratum to ``Topological properties of Wa\.zewski dendrite groups''}

\author[\initial{B.} \lastname{Duchesne}]{\firstname{Bruno} \lastname{Duchesne}}
\address{Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay,\\
Bâtiment 307, 91405, Orsay, France}
\email{bruno.duchesne@universite-paris-saclay.fr}
\urladdr{https://www.imo.universite-paris-saclay.fr/~bruno.duchesne/}

\begin{abstract}
In \cite{MR4077581}, the universal minimal flow of the group of homeomorphisms of the universal Wa\.zewski dendrite was identified as the completion of some coset space. A more concrete description was erroneously given in that paper as Basso and Tsankov explained in \cite{MR4549426}. The aim of this note is to give a corrected description of this universal minimal flow.

More Precisely, Theorem 1.12 in \cite{MR4077581} (that is Theorem 7.16 with a more precise statement in the body of the article) is wrong and should be replaced by Theorem 2 and Theorem 4 below. The mistake is located in the proof of Theorem 7.16 and the other statements are not affected by the mistake.
\end{abstract}

\subjclass{22F50, 57S05, 37B05}

\keywords{Wa\.zewski dendrites, groups of homeomorphisms, Polish groups, Steinhaus property, generic elements, automatic continuity, universal flows}

\altkeywords{Dendrites de Wa\.zewski, groupes d’homéomorphismes, groupes polonais, propriété de Steinhaus, éléments génériques, continuité automatique, flots universels}

\alttitle{Erratum à \guillemotleft\,Propriétés topologiques des groupes d’homéomorphismes des dendrites de Wa\.zewski\,\guillemotright}

\begin{altabstract}
Dans \cite{MR4077581}, le flot minimal universel du groupe des homéomorphismes de la dendrite universelle de Wa\.zewski a été identifié comme la complétion d'un certain espace de classes d'équivalence. Une description plus concrète a été donnée par erreur dans cet article, comme Basso et Tsankov l'ont expliqué dans \cite{MR4549426}. Le but de cette note est de donner une description corrigée de ce flot minimal universel.

Plus précisément, le théorème 1.12 dans \cite{MR4077581} (qui est le théorème 7.16 avec un énoncé plus précis dans le corps de l'article) est erroné et devrait être remplacé par le théorème 2 et le théorème 4 ci-dessous. L'erreur se situe dans la preuve du théorème 7.16 et les autres énoncés ne sont pas affectés par l'erreur.
\end{altabstract}

\maketitle
\vspace*{-\baselineskip}
\tableofcontents

\mainmatter

\section{Corrected statement}
We use the same notations as in \cite{MR4077581} and refer to it for the context. So $D_\infty$ is the universal Wa\.zewski dendrite, that is the unique dendrite with dense branch points set and such that any branch point has infinite order. Let $G$ be its homeomorphism group. We denote by $\CLO(D_\infty)$ the set of convex and converging linear orders on the set of branch points $\Br(D_\infty)$ \cite[Def.\,7.3]{MR4077581}. Basso and Tsankov gave a simpler definition \cite[\S 9]{MR4549426} of its elements. Let us recall it:

\begin{defn} Let $\prec$ be a linear order on the set $\Br(D_\infty)$.
\begin{itemize}
\item The order $\prec$ is \emph{converging} if for any three distinct branch points $x_1,x_2,x_3$ such that $x_1\prec x_3\prec x_2$ then $x_2$ is not between $x_1$ and $x_3$.
\item A converging order is, moreover, \emph{convex} if for any points $x_1,x_2,x_3,x_4$ such that $x_2\in(x_1,x_3)$ and $x_3\in(x_2,x_4)$ then it is not the case that $x_2\prec x_3\prec x_1\prec x_4$.
\end{itemize}

\end{defn}
Let us recall that for a converging linear order on a dendrite, any minimizing sequence has a limit and this limit does not depend on the minimizing sequence \cite[Lem.\,7.2]{MR4077581}. It is called the root of the linear order.

The space $\CLO(D_\infty)$ of convex converging linear orders is a closed $G$-invariant subspace of the space of all linear orders on all branch points and thus a $G$-flow. In~\cite[Th.\,1.12]{MR4077581}, it is stated that $\CLO(D_\infty)$ is the universal minimal flow of~$G$. This is false as Basso and Tsankov proved in \cite[Th.\,9.3]{MR4549426}. Among convex converging linear orders, there is a generic $G$-orbit. The elements of the generic orbit can be described in the following way: the root is some end point $\xi$ and for each branch point $b$, the linear order induced on branches around $b$ not containing $\xi$ is isomorphic to the standard order on rational numbers. Let us fix one of these convex converging linear orders and denote it $\prec_0$. Its stabilizer is denoted $G_{\prec_0}$.

Let $K$ be the blown-up dendrite $\overline{\Ends(D_\infty)}$, that is the completion of the space of ends of $D_\infty$ with respect to right uniformity when $\Ends(D_\infty)$ is identified with $G/G_\xi$ where $G_\xi$ is the stabilizer of an end point $\xi\in D_\infty$ in $G$. This appears to be the universal Furstenberg boundary of $G$ \cite[\S8.2]{MR4077581}.

An element $C$ of $K$ can be identified with a collection $(C_b)_{b\in\Br(D_\infty)}$ where $C_b$ is an element of $\overline{b}=\hat{b}\cup\{\infty\}$, the one-point compactification of the discrete space $\hat{b}$ of branches around $b\in\Br(D_\infty)$.

We introduce the following subset of $\CLO(D_\infty)\times K$:
\[M=\bigl\{(\prec,C)\in\CLO(D_\infty)\times K,\ \forall b,b'\in\Br(D_\infty),\ b\prec b'\implies b\in C_b'\bigr\}.\]

Since $\CLO(D_\infty)$ and $K$ are $G$-flows, it follows that $M$ is a $G$-flow as well since it is invariant and closed.

For a uniform space $X$, we denote by $\widehat{X}$ its completion. For example, $K=\widehat{G/G_\xi}$. We can now give the correct concrete identification of the universal minimal flow of~$G$.

\begin{thm}\label{th}
The universal minimal flow of $G$ is $M\simeq \widehat{G/G_{\prec_0}}$.
\end{thm}

\begin{rem}This universal minimal flow can also be recovered using highly proximal extensions. Since $G \curvearrowright \CLO(D_\infty)$ is minimal and highly proximal extensions preserve minimality \cite{MR442906}, $G \curvearrowright S_G(\CLO(D_\infty))$ (the universal highly proximal exten\-sion of $\CLO(D_\infty)$) is minimal. By \cite[Prop.\,2.7]{MR4549426}, $S_G(\CLO(D_\infty)) \simeq \Sa(G/G_{\prec_0})$ (the Samuel compactification of $G/G_{\prec_0}$), so by \cite[Prop.\,6.6]{MR4266370}, $G_{\prec_0}$ is pre-syndetic.
Since $G_{\prec_0}$ is also extremely amenable \cite[Prop.\,7.9]{MR4077581}, by \cite[Th.\,7.5]{MR4266370},
\[
M(G) \simeq \Sa(G/G_{\prec_0}) \simeq S_G(\CLO(D_\infty)).
\]
\end{rem}

The mistake in the proof of \cite[Th.\,7.16]{MR4077581} is located in the statement that the embedding $G/G_\xi\simeq \CLO(D_\infty)$ is bi-uniformly continuous. It is uniformly continuous but its inverse is not. Let us conclude with points in \cite{MR4077581} that depend on this theorem and see how they are affected.

Corollary 7.17 in \cite{MR4077581} still holds since $M$ is metrizable (and thus has a comeager orbit). The universal minimal flow of $G_\xi$ is as depicted in \cite{MR4077581}. Let us denote by $M_\xi=\{m\in M,\ \pi_1(m)=\xi\}$ where $\pi_1(\prec,C)\in D_\infty$ is the root of $\prec$.

\begin{thm} \label{umfxi}The universal minimal flow of $G_\xi$ is indeed $M_\xi\simeq \CLO(D_\infty)_\xi$.
\end{thm}

This result is used in the proof of the amenability of $G_\xi$ \cite[Th.\,8.5]{MR4077581} and this amenability result is unaffected.

\subsubsection*{Acknowledgements}
I warmly thank Gianluca Basso and Todor Tsankov for explanations about the mistake and more specifically Gianluca Basso for discussions that follow.

\section{Proofs}

\skpt
\begin{proof}[Proof of Theorem \ref{th}]
\vspace*{-.7\baselineskip}
\subsubsection*{Identification between $M$ and $\widehat{G/G_{\prec_0}}$}
For $\xi\in\Ends(D_\infty)$, we denote by $C^\xi$ the element of $K$ such that for any $b\in\Br(D_\infty)$, $C^\xi_b$ is the branch around $b$ containing $\xi$ (denoted $C_b(\xi)$ in the original paper). Let $\xi_0$ be the root of $\prec_0$ and $C^0=C^{\xi_0}$. Let us consider the orbit map $g\mapsto (g\prec_0,gC^0)$ from $G$ to $M$. On the image in $M$ of this orbit, a basis of entourages for the uniform structure is given by entourages
\[V_F=\{((\prec_1,C^{\xi_1}),(\prec_2,C^{\xi_2})\colon\ \forall x\neq y \in F,\ x\prec_1y\Leftrightarrow x\prec_2 y,\ \exists \gamma\in U_F,\ \gamma \xi_1=\xi_2\},\]
where $F$ is some finite subset of $\Br(D_\infty)$ and $U_F$ is its pointwise stabilizer. Moreover, it suffices to consider finite $c$-closed subsets $F$ since the family given by these subsets is cofinal for inclusion. The description of the uniform structure comes from the identification proved in \cite[Prop.\,8.13]{MR4077581}. Since $G_{\prec_0}$ fixes $\xi_0$ and thus $C^0$, this orbit map induces a map $G/G_{\prec_0}\to M$ that is uniformly continuous and thus extends to an equivariant continuous map $\widehat{G/G_{\prec_0}}\to M$. Let us see that the inverse map from the $G$-orbit of $(\prec_0,C^0)$ to $G/G_{\prec_0}$ is uniformly continuous as well.

The left uniform structure on the quotient space $G/G_{\prec_0}$ is generated by the system of basic entourages $E_F=\{(gG_{\prec_0},hG_{\prec_0}), g\in U_FhG_{\prec_0}\}$ where $F$ is a finite subset of $\Br(D_\infty)$ as above.

So let us fix some finite subset $F\subset\Br(D_\infty)$ and let us prove that for two elements $g,h\in G$, if $\left((g\prec_0,g\xi_0), (h\prec_0,h\xi_0)\right)\in V_F$ then $(gG_{\prec_0},hG_{\prec_0})\in E_F$. This will prove the uniform continuity of the inverse of the orbit map $G/G_{\prec_0}\to M$ on its image.

In that case by the description of $V_F$, there is $u\in U_F$ such that $g\xi_0=uh\xi_0$. Then, applying Lemma \ref{pointwise} (see below) to $\prec{}=uh\prec_0$ and $\prec'{}=g\prec_0$, there is $v\in G$ fixing pointwise $F$ and $\xi=g\xi_0$ such that $g\prec_0{}=vuh\prec_0$ and thus for $u'=vu\in U_F$ one has $g\in u'hG_{\prec_0}$ as required.

\subsubsection*{Minimality}
Let $(\prec_1,C^1),(\prec_2,C^2)\in M$. We aim to show that the closure of the $G$-orbit of $(\prec_1,C^1)$ contains $(\prec_2,C^2)$. Fix $F\subset\Br(D_\infty)$ finite and for each $b\in F$, choose a neighborhood $N_b$ of $C^2_b$. We may assume that $N_b$ is a singleton if $C^2_b\neq\infty$. To prove that the closure of the $G$-orbit of $(\prec_1,C^1)$ contains $(\prec_2,C^2)$, it suffices to prove there is $g\in G$ such that $g\prec_1$ coincides with $\prec_2$ on $F$ and $gC^1_b\in N_b$ for all $b\in F$.

Let $r_1$ be the root of $\prec_1$. Let $g\in G$ such that $gr_1\in\bigcap_{b\in F}N_b$ (this intersection is non-empty since eventually it contains a minimizing sequence for $\prec_2$). Let $b_0$ be the projection (\ie the image by the first-point map) of $gr_1$ on $[F]$. Let $g_1\in G$ fixing pointwise the branch around $b_0$ containing $gr_1$ and permuting the branches around $b_0$ containing elements of $F$ in order that the orders induced on these branches coincide for $\prec_2$ and $g_1g\prec_1$. The closure $B_1,\dots,B_n$ of each of these branches is homeomorphic to $D_\infty$ itself and $b_0$ is an end point for these subdendrites. One can now use \cite[Lem.\,7.14]{MR4077581} for each $b_i$ and find an element $h_i\in\Homeo(B_i)$ fixing $b_0$ such that $h_ig_1g\prec_1$ and $\prec_2$ coincide on $F\cap B_i$. Patching the $h_i$'s (extended by the identity on the other branches around $b_0$), one gets an element $g_2$ such that $g_2g_1g\prec_1$ and $\prec_2$ coincides on $F$. Now, it suffices to observe that for each $b\in F$, $g_2g_1gC^1_b$ is the branch around $g_2g_1gb$ containing the root of $g_2g_1g\prec_1$ that is $C^2_b$ or lies in $N_b$ if $b$ is the root of $\prec_2$.

\subsubsection*{Universality}
Since $G_{\prec_0}$ is extremely amenable, for any $G$-flow $X$, there is a $G_{\prec_0}$-fixed point $x$. The orbit map $g\mapsto gx$ is uniformly continuous and thus extends to a $G$-map $\widehat{G/G_{\prec_0}}\to X$. If $X$ is moreover minimal, the image is $X$ itself.
\end{proof}

\begin{lem}\label{pointwise} Let $\prec,\prec'$ be two convex converging linear orders with root $\xi\in\Ends(D_\infty)$ in the $G$-orbit of $\prec_0$ and $F$ be a finite $c$-closed subset of $\Br(D_\infty)$ such that for any $x,y\in F$, $x\prec y\Leftrightarrow x\prec'y$. Then there exists $v\in G_\xi$ fixing pointwise $F$ such that $v\prec{}={}\prec'$.
\end{lem}

\begin{proof} First, we may assume that $F$ contains the projection $p$ of $\xi$ to the subdendrite generated by $F$. Actually if this point $p$ does not belong to $F$, we observe that since these orders are converging and they have the same root, $p$ has to be smaller than any other element of $F$ for $\prec$ and $\prec'$.

We use a back and forth argument to construct $v$. Actually, we construct $v$ as a bijection of $\Br(D_\infty)$ preserving the betweenness relation and use the identification of~$G$ with the group of betweenness preserving bijections of $\Br(D_\infty)$.

Let $\{x_n\}_{n\geq1}$ be an enumeration of $\Br(D_\infty)\setminus F$. We construct increasing sequences $X_n,Y_n$ of finite subsets of $\Br(D_\infty)$ containing $F$ and bijections $v_n\colon X_n\to Y_n$ such that for any $n\in\mathbb{N}$:

\begin{enumerate}
\item $X_n, Y_n$ are $c$-closed and contain the projection of $\xi$ on the subdendrite they generate.
\item $x_n\in X_n\cap Y_n$,
\item $v_n$ preserves the betweenness relation and for any $x,y\in X_n$,
\[
x\prec y\iff v_n(x)\prec'v_n(y),
\]
\item if $p_n\in X_n$ is the projection of $\xi$ on the subdendrite generated by $X_n$ then $v_n(p_n)$ is the projection of $\xi$ on the subdendrite generated by $Y_n$,
\item $v_{n+1}|_{X_n}=v_n$.
\end{enumerate}

We set $X_0=Y_0=F$ and $v_0$ to be the identity on $F$. We proceed by induction. Assume for some $n\in\mathbb{N}$, $X_n,Y_n$ and $v_n$ have been defined and satisfy the above properties. Let $c_n$ be the projection of $x_{n+1}$ on the subdendrite generated by \hbox{$X_n\cup\{\xi\}$}. Let $\left[a_n,b_n\right]$ be the minimal arc (maybe reduced to $c_n$ if $c_n\in X_n$) such that $a_n,b_n\in X_n\cup\{\xi\}$ containing $c_n$. Let $c'_n$ be a branch point in $\left(v_n(a_n),v_n(b_n)\right)$ (with the convention that $v_n(\xi)=\xi$) or $c'_n=v_n(a_n)=v_n(b_n)$ if $c_n\in X_n$.

Let us observe that for any $f\in X_n$, $c_n\prec f\Leftrightarrow c'_n\prec' v_n(f)$. Actually, let $d_n=c(\xi,c_n,f)\in X_n$, if $d_n=c_n$ then $c_n\prec d_n$, if $d_n=f$ then $f\prec c_n$ (by the converging property) and otherwise
\[
f\prec c_n\iff f\prec a_n \iff f\prec b_n
\]
(see \cite[Lem.\,7.5]{MR4077581}). These properties are true because $c_n\in [a_n,b_n]$ and thus, the same holds for any $c'_n\in[v_n(a_n),v_n(b_n)]$.

If $x_n=c_n$ then let $x'_n$ be $c'_n$ and otherwise choose a point $x'_{n}$ in some component not containing $\xi, v_n(a_n), v_n(b_n)$ around $c'_n$ such that for any $x\in X_n$,
\[
x_{n+1}\prec x\iff x'_n\prec'v_n(x).
\]
This is possible because $\prec'$ is in the $G$-orbit of $\prec_0$ and thus the linear order induced by $\prec'$ on branches around $c'_n$ not containing $\xi$ is isomorphic to the standard order on rational numbers. Let $Y_n'=Y_n\cup\{x'_n,c'_n\}$ and $X_n'=X_n\cup\{x_{n+1},c_n\}$ and let us extend $v_n$ on $X_n'$ with $v_n(x_{n+1})=x'_n$ and $v_n(c_n)=c'_n$.

Similarly let $d'_n$ be the projection of $x_{n+1}$ on the subdendrite generated by \hbox{$Y'_n\cup\{\xi\}$}. If $d'_n\notin Y'_n$, let $\left[a'_n,b'_n\right]$ be the minimal arc such that $a'_n,b'_n\in Y'_n\cup\{\xi\}$ and containing~$d'_n$. Let $d_n$ be a branch point in $\left(v_n^{-1}(a'_n),v_n^{-1}(b'_n)\right)$ and choose a point~$y_n$ in some component around $d_n$ not intersecting $X'_n\cup\{\xi\}$ and such that for any $x\in X_n$,
\[
y_n\prec x\iff x_{n+1}\prec'v_n(x).
\]

Now let $X_{n+1}=X'_n\cup\{d_n,y_n\}$, $Y_{n+1}=Y'_n\cup\{d'_n,x_{n+1}\}$ and $v_{n+1}(x)=v_n(x)$ for all $x\in X'_n$, $v_{n+1}(d_n)=d'_{n}$ and $v_{n+1}(y_n)=x_{n+1}$ (if these new points do not already belong to $X'_n$).

By construction, $X_n,Y_n$ and $v_n$ have the announced properties and finally, for $x\in \Br(D_\infty)$ we define $v(x)=v_n(x)$ for any $n$ large enough such that $x\in X_n$. The map $v$ is a bijection that preserves the betweenness relation and maps $\prec$ to $\prec'$. Because of the fourth property, $v$ fixes $\xi$. Thus $v$ fixes pointwise $F\cup\{\xi\}$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{umfxi}] Since $G$ acts transitively on $\Ends(D_\infty)$, we continue with $\xi=\nobreak\xi_0$ the root of $\prec_0$. The space $M_{\xi_0}$ is closed and thus compact since $\pi_1$ is continuous. Let $\pi$ be the projection $M\to\CLO(D_\infty)$. This map is continuous on $M$ and injective on $M_{\xi_0}$ (in that case, the second component is constant equal to $C^0$) with image $\CLO(D_\infty)_{\xi_0}$. Since $\CLO(D_\infty)_{\xi_0}$ and $M_{\xi_0}$ are both compact, these two spaces are homeomorphic as $G_{\xi_0}$-flows. The first part of the proof of Theorem~\ref{th} also proves that $G_{\xi_0}/G_{\prec_0}$ is bi-uniformly embedded in $M_{\xi_0}$ and since it is dense $\widehat{G_{\xi_0}/G_{\prec_0}}\simeq M_{\xi_0}$. Minimality and universality go as in the proof of Theorem~\ref{th}.
\end{proof}

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