$\varphi(x)c^{\pi_{1}^{\prime}}=\varphi(xc^{\pi_{1}})\in\operatorname{\mathrm{dom}}(\tau^{\prime})$,
so that in $\mathcal{G}_{\tau^{\prime}}$:

$\mathcal{G}_{(\varphi_{1},\varphi_{2})}(e)=\varphi_{1}(xc^{\pi_{1}})\Sigma_{1}=\varphi_{1}(x)c^{\pi^{\prime}_{1}}\Sigma_{1}=e_{1,x\varphi_{1}}(c)\,.$

In other words, the restriction of $\mathcal{G}_{(\varphi_{1},\varphi_{2})}$
to the star at $x\Gamma_{1}$ is the composition $e_{1,\varphi_{1}(x)}\circ
e_{1,x}^{-1}$. Furthermore, this map is an injection into the star at
$\varphi_{1}(x)\Gamma_{1}$.

Similarly, one can prove that the restriction of
$\mathcal{G}_{(\varphi_{1},\varphi_{2})}$ to the star at a vertex
$y\Gamma_{2}$ is the composition $e_{2,\varphi_{2}(y)}\circ e_{2,y}^{-1}$,
which is an injection into the star at $\varphi_{2}(y)\Gamma_{2}$. ∎

###  5.3. Paths in Bass-Serre graphs of global pre-actions

Let us turn to the case of a global pre-action $(X_{1},X_{2},\tau)$. In this
case, the bijections $e_{1,x}$ and $e_{2,y}$, defined at the end of Section
5.1, become just

$e_{1,x}:\,C_{1}\longrightarrow\operatorname{\mathrm{st}}(x\Gamma_{1})\quad\text{
and }\quad
e_{2,y}:\,C_{2}\longrightarrow\operatorname{\mathrm{st}}(y\Gamma_{2})\,.$

Given a point $x\in X_{1}$ and an element $\gamma\in N_{C_{2}}$ with normal
form $\gamma=c_{1}\cdots c_{n}\sigma$ where $n\geq 1$ and $c_{1}\in
C_{2}-\\{1\\}$, we associate a sequence $(x_{0},x_{1},\ldots,x_{n+1})$ in
$X_{1}\cup X_{2}$ and a sequence $(e_{0},e_{1},...,e_{n})$ of edges in the
Bass-Serre graph as follows. We set $x_{0}=x$, $c_{0}=1\in C_{1}$, and then
inductively for $i=0,\ldots,n$:

  * •

for $i$ such that $c_{i}\in C_{1}$, set $e_{i}=e_{1,x_{i}}(c_{i})$, and
$x_{i+1}=x_{i}c_{i}\tau$;

  * •

for $i$ such that $c_{i}\in C_{2}$, set $e_{i}=e_{2,x_{i}}(c_{i})$, and
$x_{i+1}=x_{i}c_{i}\tau^{-1}$.

Notice that, for any $i=0,\ldots,n-1$, if $c_{i}\in C_{1}$ (or equivalently if
$i$ is even), one has
$r(e_{i})=x_{i}c_{i}\tau\Gamma_{2}=x_{i+1}\Gamma_{2}=s(e_{i+1})$ , and
similarly if $c_{i}\in C_{2}$ we have $r(e_{i})=s(e_{i+1})$. Hence
$(e_{0},\ldots,e_{n})$ is a path, that we denote by
$\mathsf{path}_{1,x}(\gamma)$. Note that this path begins by the edge
$e_{0}=x\Sigma_{1}$.

Let us check that $\mathsf{path}_{1,x}(\gamma)$ is a reduced path. For $0\leq
i\leq n-1$ and $c_{i}\in C_{1}$, we have

$e_{i+1}=\bar{e}_{i}\Leftrightarrow
x_{i+1}c_{i+1}\Sigma_{2}=\overline{x_{i}c_{i}\Sigma_{1}}\Leftrightarrow
x_{i}c_{i}\tau c_{i+1}\Sigma_{2}=x_{i}c_{i}\tau\Sigma_{2}\Leftrightarrow
c_{i+1}=1$

since $X_{2}\curvearrowleft^{\pi_{2}}\Gamma_{2}$ is free. Since $c_{1}\cdots
c_{n}\sigma$ is the normal form of $\gamma$, we cannot have $c_{i+1}=1$, so
$\mathsf{path}_{1,x}(\gamma)$ is reduced.

Finally, since the maps $e_{1,x}$ and $e_{2,x}$ are bijective, given a reduced
path $(e_{0},\ldots,e_{n})$ beginning by $x\Sigma_{1}$, there is exactly one
normal form $c_{1}\cdots c_{n}$ with $c_{1}\in C_{2}-\\{1\\}$ such that
$\mathsf{path}_{1,x}(c_{1}\cdots c_{n})=(e_{0},\ldots,e_{n})$. The following
remark is now clear.

######  Remark 5.10.

For any $x\in X_{1}$, the map $\mathsf{path}_{1,x}$ is a surjection from
$N_{C_{2}}$ to the set of reduced paths starting by the edge $x\Sigma_{1}$. It
becomes a bijection if we restrict it to the subset of elements $\gamma\in
N_{C_{2}}$ whose normal form is $c_{1}\cdots c_{n}$ with $c_{1}\in
C_{2}-\\{1\\}$. So if $x\Sigma_{1}$ is a treeing edge, then the images
$\mathsf{path}_{1,x}(\gamma)$, for $\gamma\in N_{C_{2}}$, cover exactly the
half-tree of $x\Sigma_{1}$ in $\mathcal{G}_{\tau}$.

We now give a definition of path-type elements which is analogous to the one
for HNN extensions, except that we only want to consider paths which end in
$X_{1}$. An element $\gamma\in N_{C_{2}}$ with normal form $\gamma=c_{1}\cdots
c_{n}$ such that $c_{1}\in C_{2}-\\{1\\}$ and $n\geq 1$ is _odd_ will be
called a path type element of $N_{C_{2}}$. Note that the corresponding path
then has _even_ length. If $\gamma^{\prime}=c_{1}\cdots c_{k}$, for some
$k\leq n$, is a path type element in $N_{C_{2}}$, then $\gamma$ is called a
path type extension of $\gamma^{\prime}$.

Similarly, given a point $x\in X_{2}$, we can associate to every element
$\gamma\in N_{C_{1}}$ with normal form $\gamma=c_{1}\cdots c_{n}\sigma$, where
$n\geq 1$ and $c_{1}\in C_{1}-\\{1\\}$, a reduced path
$\mathsf{path}_{2,x}(\gamma):=(e_{0},e_{1},...,e_{n})$, as follows. We set
$x_{0}=x$, $s_{0}=1\in C_{2}$, and then inductively for $i=0,\ldots,n$:

  * •

for $i$ such that $c_{i}\in C_{1}$, set $e_{i}=e_{1,x_{i}}(s_{i})$, and
$x_{i+1}=x_{i}c_{i}\tau$;

  * •

for $i$ such that $c_{i}\in C_{2}$, set $e_{i}=e_{2,x_{i}}(s_{i})$, and
$x_{i+1}=x_{i}c_{i}\tau^{-1}$.

This defines a surjective map $\mathsf{path}_{2,x}$ from $N_{C_{1}}$ to the
set of reduced paths starting by the edge $x\Sigma_{2}$. Hence, if
$x\Sigma_{2}$ is a treeing edge then, the images
$\mathsf{path}_{2,x}(\gamma)$, for $\gamma\in N_{C_{1}}$, cover exactly the
half-tree of $x\Sigma_{2}$ in $\mathcal{G}_{\tau}$. Moreover, the map
$\mathsf{path}_{2,x}$ becomes a bijection if we restrict it to the set of
elements of with normal form $\gamma=c_{1}\cdots c_{n}$ where $n\geq 1$ and
$c_{1}\in C_{1}-\\{1\\}$, and such elements will be called path type elements
of $N_{C_{1}}$ when moreover $n$ is odd. As before, there is a notion of path
type extension for path type elements in $N_{C_{1}}$.

######  Remark 5.11.

Let $i,j\in\\{1,2\\}$ with $i\neq j$ and $x\in X_{j}$, $\gamma\in N_{C_{i}}$
with normal form $c_{1}\cdots c_{n}\sigma$.

  1. (1)

By construction, $\mathsf{path}_{j,x}(\gamma)=\mathsf{path}_{j,x}(c_{1}\cdots
c_{n})$ and, for every $1\leq k\leq n$, the path
$\mathsf{path}_{j,x}(c_{1}\cdots c_{n})$ is an extension of
$\mathsf{path}_{j,x}(c_{1}\cdots c_{k})$.

  2. (2)

The source of $\mathsf{path}_{j,x}(\gamma)$ is $s(x\Sigma_{j})=x\Gamma_{j}$.

  3. (3)

If $\gamma=c_{1}\cdots c_{n}$ is a path type element, then the range of
$\mathsf{path}_{j,x}(\gamma)$ is $x\gamma^{\pi_{j,\tau}}\Gamma_{j}$.

  4. (4)

If for some $1\leq k\leq n$ the last edge of $\mathsf{path}_{j,x}(c_{1}\cdots
c_{k})$ is a treeing edge, then for all $k\leq l\leq n$ the last edge of
$\mathsf{path}_{j,x}(c_{1}\cdots c_{l})$ is also a treeing edge.

Let us end this section by establishing a link between paths in Bass-Serre
trees and Bass-Serre graphs.

######  Remark 5.12.

Consider a global pre-action $(X_{1},X_{2},\tau)$, and basepoints $x_{1}\in
X_{1}$ and $x_{2}\in X_{2}$ such that $x_{2}=x_{1}\tau$. There exists a unique
morphism of pre-actions

$(\varphi_{1},\varphi_{2}):(\Gamma,\Gamma,\mathord{\operatorname{id}})\to(X_{1},X_{2},\tau)$

from the translation pre-action, such that $\varphi_{j}(1)=x_{j}$ for $j=1,2$.
It satisfies $\varphi_{j}(\gamma)=x_{j}\gamma^{\pi_{j,\tau}}$ for all $j=1,2$
and $\gamma\in\Gamma$. By restriction, one obtains morphisms

$\displaystyle(\varphi_{1,+},\varphi_{2,+})$ $\displaystyle:(\Gamma_{1}\sqcup
N_{C_{2}},\Gamma_{2}\cup N_{C_{2}},\tau_{+})\to(X_{1},X_{2},\tau)$
$\displaystyle(\varphi_{1,-},\varphi_{2,-})$ $\displaystyle:(\Gamma_{1}\cup
N_{C_{1}},\Gamma_{2}\sqcup N_{C_{1}},\tau_{-})\to(X_{1},X_{2},\tau)$

from the positive and negative translation pre-actions.

######  Lemma 5.13.

In the context of the above remark, the Bass-Serre morphism
$\mathbf{BS}(\varphi_{1},\varphi_{2})$, from the Bass-Serre tree $\mathcal{T}$
to the Bass-Serre graph $\mathcal{G}_{\tau}$, sends
$\mathsf{path}_{j,1_{\Gamma}}^{\mathcal{T}}(\gamma)$ onto
$\mathsf{path}_{j,x_{j}}^{\mathcal{G}_{\tau}}(\gamma)$ for all $j\in\\{1,2\\}$
and all $\gamma\in N_{C_{i}}$ with $i\in\\{1,2\\}-\\{j\\}$.

###### Proof.

We make the proof in the case $j=1$ only; the case $j=2$ is similar.

Let us consider $\gamma\in N_{C_{2}}$, and write its normal form:
$\gamma=c_{1}t^{\varepsilon_{1}}\cdots c_{n}t^{\varepsilon_{n}}h_{n+1}$. Let
us denote by $(e_{0},e_{1},\ldots,e_{n})$ the edges of
$\mathsf{path}_{1,1_{\Gamma}}^{\mathcal{T}}(\gamma)$, and by
$(e^{\prime}_{0},e^{\prime}_{1},\ldots,e^{\prime}_{n})$ the edges of
$\mathsf{path}_{1,x_{1}}^{\mathcal{G}_{\tau}}(\gamma)$. The auxiliary
sequences in $\Gamma$ and $X$ used in the construction of the paths will be
denoted by $(\gamma_{0},\ldots,\gamma_{n+1})$ and $(x_{0},\ldots,x_{n+1})$
respectively.

An easy induction shows that $x_{i}=\varphi_{1}(\gamma_{i})$ when $i$ even,
and $x_{i}=\varphi_{2}(\gamma_{i})$ when $i$ is odd. Then, we notice that the
source of $e_{i}=e_{1,\gamma_{i}}(c_{i})$ is $\gamma_{i}\Gamma_{1}$ when $i$
even, and the source of $e_{i}=e_{2,\gamma_{i}}(c_{i})$ is
$\gamma_{i}\Gamma_{2}$ when $i$ odd. Thus, using Lemma 5.9, we get

$\mathbf{BS}(\varphi_{1},\varphi_{2})(e_{i})=e_{1,\varphi_{1}(\gamma_{i})}\circ
e_{1,\gamma_{i}}^{-1}\big{(}e_{1,\gamma_{i}}(c_{i})\big{)}=e_{1,x_{i}}(c_{i})=e^{\prime}_{i}$

when $i$ is even, and

$\mathbf{BS}(\varphi_{1},\varphi_{2})(e_{i})=e_{2,\varphi_{2}(\gamma_{i})}\circ
e_{2,\gamma_{i}}^{-1}\big{(}e_{2,\gamma_{i}}(c_{i})\big{)}=e_{2,x_{i}}(c_{i})=e^{\prime}_{i}$

when $i$ is odd. ∎

Therefore, if $x_{1}\Sigma_{1}$ is a treeing edge then, the image of
$\mathbf{BS}(\varphi_{1,+},\varphi_{2,+})$ is the half-tree of
$x_{1}\Sigma_{1}$ while, if $x_{2}\Sigma_{2}$ is a treeing edge, the image of
$\mathbf{BS}(\varphi_{1,-},\varphi_{2,-})$ is the half-tree of
$x_{2}\Sigma_{2}$.

###  5.4. The free globalization of a pre-action of an amalgam

First, let us notice that, for any $\sigma\in\Sigma$, there is an automorphism
of pre-actions induced by left translation by $\sigma$

$(\gamma\mapsto\sigma\gamma,\gamma\mapsto\sigma\gamma)$

for each of the following pre-actions:

  * •

the translation pre-action $(\Gamma,\Gamma,\mathord{\operatorname{id}})$;

  * •

the positive translation pre-action $(\Gamma_{1}\sqcup
N_{C_{2}},\Gamma_{2}\cup N_{C_{2}},\tau_{+})$;

  * •

the negative translation pre-action $(\Gamma_{1}\cup
N_{C_{1}},\Gamma_{2}\sqcup N_{C_{1}},\tau_{-})$.

Indeed, all sets $\Gamma,\Gamma_{1},\Gamma_{2},\Sigma,N_{C_{1}},N_{C_{2}}$ are
invariant by left translation by $\sigma$, hence the domains and range of
$\tau_{+}$ and $\tau_{-}$ are invariant by left translation by $\sigma$. Then
checking we have morphisms of pre-actions is a straightforward computation,
and invertibility is obvious.

######  Proposition 5.14.

Consider a global pre-action $(X_{1},X_{2},\tau)$, and basepoints $x_{1}\in
X_{1}$ and $x_{2}=x_{1}\tau\in X_{2}$. The following are equivalent:

  1. (i)

the morphism of pre-actions $(\varphi_{1,+},\varphi_{2,+}):(\Gamma_{1}\sqcup
N_{C_{2}},\Gamma_{2}\cup N_{C_{2}},\tau_{+})\to(X_{1},X_{2},\tau)$ of Remark
5.12 is injective;

  2. (ii)

the morphism of graphs $\mathbf{BS}(\varphi_{1,+},\varphi_{2,+})$ is
injective;

  3. (iii)

the edge $x_{1}\Sigma_{1}$ in the Bass-Serre graph
$\mathbf{BS}(X_{1},X_{2},\tau)$ is a treeing edge.

###### Proof.

For all $\gamma\in\Gamma_{j}\cup N_{C_{2}}$, recall that
$\varphi_{j,+}(\gamma)=x_{j}\gamma^{\pi_{j,\tau}}$, so that
$\mathbf{BS}(\varphi_{1,+},\varphi_{2,+})$ sends vertices $\gamma\Gamma_{j}$
to $x_{j}\gamma^{\pi_{j,\tau}}\Gamma_{j}$, and edges $\gamma\Sigma_{j}$ to
$x_{j}\gamma^{\pi_{j,\tau}}\Sigma_{j}$. Fixing $\gamma$, we get
$\varphi_{j,+}(\gamma g)=x_{j}\gamma^{\pi_{j,\tau}}g^{\pi_{j}}$ for
$g\in\Gamma_{j}$; since $X_{j}\curvearrowleft^{\pi_{j}}\Gamma_{j}$ is free,
$\varphi_{j,+}$ realizes a bijection between $\gamma\Gamma_{j}$ and
$x_{j}\gamma^{\pi_{j,\tau}}\Gamma_{j}$, and also a bijection between
$\gamma\Sigma_{j}$ and $x_{j}\gamma^{\pi_{j,\tau}}\Sigma_{j}$. Consequently,
$\varphi_{j,+}$ is injective if an only if $\gamma\Gamma_{j}\mapsto
x_{j}\gamma^{\pi_{j,\tau}}\Gamma_{j}$ and $\gamma\Sigma_{j}\mapsto
x_{j}\gamma^{\pi_{j,\tau}}\Sigma_{j}$ are both injective. This proves that (i)
and (ii) are equivalent.

The implication (iii) $\implies$ (ii) follows from the fact that when
$x_{1}\Sigma_{1}$ is a treeing edge $\mathbf{BS}(\varphi_{1,+},\varphi_{2,+})$
is locally injective from the half-tree of $\Sigma_{1}$ to the half-tree of
$x_{1}\Sigma_{1}$, hence $\mathbf{BS}(\varphi_{1,+},\varphi_{2,+})$ is
injective.

Finally assume (ii) and let $\omega$ be a reduced path starting by the edge
$x_{1}\Sigma_{1}$. By Remark 5.10 there exists $\gamma\in N_{C_{2}}$ such that
$\omega=\mathsf{path}_{1,x_{1}}(\gamma)$. By Lemma 5.13, $\omega$ is the image
by $\mathbf{BS}(\varphi_{1,+},\varphi_{2,+})$ of
$\mathsf{path}_{1,1_{\Gamma}}^{\mathcal{T}}(\gamma)$. Since
$\mathbf{BS}(\varphi_{1,+},\varphi_{2,+})$ is supposed to be injective and
since the last vertex of $\mathsf{path}_{1,1_{\Gamma}}^{\mathcal{T}}(\gamma)$
is not $\Gamma_{1}$, we deduce that the last vertex of $\omega$ is not
$x_{1}\Gamma_{1}$. Hence, $x_{1}\Sigma_{1}$ is a treeing edge by Lemma 2.16.∎

By a very similar argument, we get also the following result.

######  Proposition 5.15.

Consider a global pre-action $(X_{1},X_{2},\tau)$, and basepoints $x_{1}\in
X_{1}$ and $x_{2}=x_{1}\tau\in X_{2}$. The following are equivalent:

  1. (i)

the morphism of pre-actions $(\varphi_{1,-},\varphi_{2,-}):(\Gamma_{1}\cup
N_{C_{1}},\Gamma_{2}\sqcup N_{C_{1}},\tau_{-})\to(X_{1},X_{2},\tau)$ of Remark
5.12 is injective;

  2. (ii)

the morphism of graphs $\mathbf{BS}(\varphi_{1,-},\varphi_{2,-})$ is
injective;

  3. (iii)

the edge $x_{2}\Sigma_{2}$ in the Bass-Serre graph
$\mathbf{BS}(X_{1},X_{2},\tau)$ is a treeing edge.

######  Remark 5.16.

Putting the two previous propositions together, one can show that, given a
global pre-action of $\Gamma$, its Bass-Serre graph is a forest if and only if
the action $X_{1}\curvearrowleft^{\pi_{1,\tau}}\Gamma$ (or equivalently
$X_{2}\curvearrowleft^{\pi_{2,\tau}}\Gamma$) is free.

Say that a pre-action is transitive when its Bass-Serre graph is connected.
Note that a global pre-action $(X_{1},X_{2},\tau)$ is transitive if and only
if the action $X_{1}\curvearrowleft^{\pi_{1,\tau}}\Gamma$, or equivalently
$X_{2}\curvearrowleft^{\pi_{2,\tau}}\Gamma$, is a transitive action. We will
show that every transitive pre-action has a canonical extension to a
transitive action, which is as free as possible. The construction is again
better described in terms of the Bass-Serre graph: we are going to attach as
many treeing edges as possible to it.

######  Theorem 5.17.

Every transitive $\Gamma$-pre-action $(X_{1},X_{2},\tau)$ admits a transitive
and global extension $(\tilde{X}_{1},\tilde{X}_{2},\tilde{\tau})$ which
satisfies the following universal property: given any transitive and global
extension $(Y_{1},Y_{2},\tau^{\prime})$ of $(X_{1},X_{2},\tau)$, there exists
a unique morphism of pre-actions
$(\varphi_{1},\varphi_{2}):(\tilde{X}_{1},\tilde{X}_{2},\tilde{\tau})\to(Y_{1},Y_{2},\tau^{\prime})$
such that

$(\varphi_{1\restriction X_{1}},\varphi_{2\restriction
X_{2}})=(\mathord{\operatorname{id}}_{X_{1}},\mathord{\operatorname{id}}_{X_{2}}).$

Moreover, all the (oriented) edges from the Bass-Serre graph
$\mathbf{BS}(X_{1},X_{2},\tau)$ to its complement in
$\mathbf{BS}(\tilde{X}_{1},\tilde{X}_{2},\tilde{\tau})$ are treeing edges.

###### Proof.

We will obtain the Bass-Serre graph of this action by adding only treeing
edges to the Bass-Serre graph of the pre-action. First enumerate the
$\Sigma_{1}$-orbits which do not belong to the domain of $\tau$ as
$(x_{i}\Sigma_{1})_{i\in I_{+}}$, and the $\Sigma_{2}$-orbits which do not
belong to the range of $\tau$ as $(x_{i}\Sigma_{2})_{i\in I_{-}}$, with
disjoint index sets $I_{+},I_{-}$. Then, we take copies
$(Y_{1,i},Y_{2,i},\tau_{i})$, of the positive translation pre-action
$(\Gamma_{1}\sqcup N_{C_{2}},\Gamma_{2}\cup N_{C_{2}},\tau_{+})$, for $i\in
I_{+}$, and copies $(Y_{1,i},Y_{2,i},\tau_{i})$ , of the negative translation
pre-action $(\Gamma_{1}\cup N_{C_{1}},\Gamma_{2}\sqcup N_{C_{1}},\tau_{-})$,
for $i\in I_{-}$, which are pairwise disjoint (by this, we mean $Y_{1,i}\cup
Y_{2,i}$ is disjoint from $Y_{1,i^{\prime}}\cup Y_{2,i^{\prime}}$ whenever
$i\neq i^{\prime}$), and disjoint from the original pre-action
$(X_{1},X_{2},\tau)$. We set then

$\tilde{X}_{1}={\left(X_{1}\sqcup\bigsqcup_{i\in
I_{+}}Y_{1,i}\sqcup\bigsqcup_{i\in
I_{-}}Y_{1,i}\right)}\bigg{/}_{\sim_{1}}\quad\text{ and
}\quad\tilde{X}_{2}={\left(X_{2}\sqcup\bigsqcup_{i\in
I_{+}}Y_{2,i}\sqcup\bigsqcup_{i\in I_{-}}Y_{2,i}\right)}\bigg{/}_{\sim_{2}}$

where $\sim_{1}$ identifies the element $x_{i}g\in X_{1}$ with
$g\in\Gamma_{1}\subset Y_{1,i}$, for each $i\in I_{+}$ and $g\in\Gamma_{1}$,
and $\sim_{2}$ identifies the element $x_{i}h\in X_{2}$ with
$h\in\Gamma_{2}\subset Y_{2,i}$, for each $i\in I_{-}$ and $h\in\Gamma_{1}$.
Since the identifications just glue some orbits pointwise and respect the
$\Gamma_{j}$-actions, $\tilde{X}_{1}$ is endowed with a free
$\Gamma_{1}$-action, and $\tilde{X}_{2}$ is endowed with a free
$\Gamma_{2}$-action. Now, we set

$\tilde{\tau}=\tau\sqcup\bigsqcup_{i\in I_{+}}\tau_{i}\sqcup\bigsqcup_{i\in
I_{-}}\tau_{i}\,,$

which is possible since the domain of $\tau_{i}$, for $i\in I_{+}$, intersects
other components in $\tilde{X}_{1}$ only in the orbit $x_{i}\Sigma_{1}$, the
range of $\tau_{i}$, for $i\in I_{+}$, does not intersect other components in
$\tilde{X}_{1}$, and the situation is analogue for $\tau_{i}$ with $i\in
I_{-}$. We have got a pre-action $(\tilde{X}_{1},\tilde{X}_{2},\tilde{\tau})$.

This pre-action is transitive, since all pre-actions $(X_{1},X_{2},\tau)$ and
$(Y_{1,i},Y_{2,i},\tau_{i})$ are, and the identifications make connections
between each $(Y_{1,i},Y_{2,i},\tau_{i})$ and $(X_{1},X_{2},\tau)$ in the
Bass-Serre graph. It is also global, since every $\Sigma_{1}$-orbit in
$Y_{1,i}$, respectively $\Sigma_{2}$-orbit in $Y_{2,i}$, which is not in the
domain, respectively the range, of $\tau_{i}$ has been identified with an
orbit in $X_{1}$, respectively $X_{2}$, and every $\Sigma_{1}$-orbit in
$X_{1}$, respectively $\Sigma_{2}$-orbit in $X_{2}$, is now in the domain,
respectively the range, of $\tilde{\tau}$.

Moreover, the (oriented) edges from the Bass-Serre graph
$\mathbf{BS}(X_{1},X_{2},\tau)$ to its complement in
$\mathbf{BS}(\tilde{X}_{1},\tilde{X}_{2},\tilde{\tau})$ are exactly the edges
$x_{i}\Sigma_{1}$ for $i\in I_{+}$, and the edges $x_{i}\Sigma_{2}$ for $i\in
I_{-}$. For each $i\in I_{+}$, the morphism of pre-actions
$(\varphi_{1,+},\varphi_{2,+}):(\Gamma_{1}\sqcup N_{C_{2}},\Gamma_{2}\cup
N_{C_{2}},\tau_{+})\to(X_{1},X_{2},\tau)$ of Remark 5.12, with basepoints
$x_{i}\in\tilde{X}_{1}$ and $x_{i}\tilde{\tau}\in\tilde{X}_{2}$, is injective
since it realizes an isomorphism onto $(Y_{1,i},Y_{2,i},\tau_{i})$, hence
$x_{i}\Sigma_{1}$ is a treeing edge by Proposition 5.14. One proves similarly
that the edges $x_{i}\Sigma_{2}$ are treeing edges using Proposition 5.15.

It now remains to prove the universal property. To do so, take any transitive
and global extension $(Y_{1},Y_{2},\tau^{\prime})$ of $(X_{1},X_{2},\tau)$.
Then, the unique morphism of pre-actions $(\varphi_{1},\varphi_{2})$ from
$(\tilde{X}_{1},\tilde{X}_{2},\tilde{\tau})$ to $(Y_{1},Y_{2},\tau^{\prime})$
such that $(\varphi_{1\restriction X_{1}},\varphi_{2\restriction
X_{2}})=(\mathord{\operatorname{id}}_{X_{1}},\mathord{\operatorname{id}}_{X_{2}})$
is obtained by taking the union of
$(\mathord{\operatorname{id}}_{X_{1}},\mathord{\operatorname{id}}_{X_{2}})$
with the morphisms $(\varphi_{1,i},\varphi_{2,i})$ from
$(Y_{1,i},Y_{2,i},\tau_{i})$ to $(Y_{1},Y_{2},\tau^{\prime})$ coming from
Remark 5.12 with respect to basepoints $x_{i}$ and $x_{i}\tilde{\tau}^{\pm
1}$, which are unique. ∎

It is straightforward to deduce from the universal property above that the
action we just built is unique up to isomorphism. We thus call it the free
globalization of the pre-action $(X_{1},X_{2},\tau)$. The interested reader
can establish a connection with the notion of partial action, as we did in
section 3.5 for HNN extensions. For the sake of brevity, we just observe the
following useful analogue of Proposition 3.27.

######  Remark 5.18.

In the context of Theorem 5.17, if the pre-action $(X_{1},X_{2},\tau)$ is not
global, then the conjugate actions $\pi_{1,\tilde{\tau}}$ and
$\pi_{2,\tilde{\tau}}$ induced by the free globalization
$(\tilde{X}_{1},\tilde{X}_{2},\tilde{\tau})$ are highly faithful. Indeed, by
Corollary 2.6, it suffices to prove that $\pi_{1,\tilde{\tau}}$ is strongly
faithful. Notice that $(\tilde{X}_{1},\tilde{X}_{2},\tilde{\tau})$ contains a
copy of the positive (or of the negative) translation pre-action, which
correspond to a half-tree in
$\mathbf{BS}(\tilde{X}_{1},\tilde{X}_{2},\tilde{\tau})$. Now note that the
positive translation pre-action is strongly faithful, meaning that given
$F\Subset\Gamma$, we can find $x\in\Gamma_{1}\cup N_{C_{1}}$ such that for all
$f\in F$, we have $xf\neq x$ and $xf\in\Gamma_{1}\cup N_{C_{1}}$ (indeed it
suffices to take $x\in N_{C_{1}}$ with a sufficiently long normal form).
Similarly, the negative translation is strongly faithful. It follows that the
free globalization is strongly faithful, hence highly faithful as wanted.

Let us furthermore observe that we can always build the free globalization on
a fixed couple of sets $(\bar{X}_{1},\bar{X}_{2})$ with $\bar{X}_{j}$
containing $X_{i}$, provided $\bar{X}_{j}$ contains infinitely many free
$\Gamma_{j}$-orbits.

######  Theorem 5.19.

Let $\bar{X}_{j}$ be a countable set equipped with a free $\Gamma_{j}$-action
for $j=1,2$. Suppose $X_{j}\subseteq\bar{X}_{j}$ is $\Gamma_{j}$-invariant,
and $\bar{X}_{j}\setminus X_{j}$ contains infinitely many $\Gamma_{j}$-orbits.
Suppose further that we have a pre-action $(X_{1},X_{2},\tau)$. Then there is
a bijection $\bar{\tau}:\bar{X}_{1}\to\bar{X}_{2}$ which extends $\tau$ such
that $(\bar{X}_{1},\bar{X}_{2},\bar{\tau})$ is (isomorphic to) the free
globalization of $(X_{1},X_{2},\tau)$.

###### Proof.

Let $(\tilde{X}_{1},\tilde{X}_{2},\tilde{\tau})$ be the free globalization of
$(X_{1},X_{2},\tau)$. The fact that $\tilde{X}_{j}\setminus X_{j}$ contains
infinitely many $\Gamma_{j}$-orbits and is countable implies that there exist
$\Gamma_{j}$-equivariant bijections $\varphi_{j}:\tilde{X}_{j}\to\bar{X}_{j}$
whose restrictions to $X_{j}$ are the identities. Then, one can push forward
the bijection $\tilde{\tau}$, to obtain a bijection
$\bar{\tau}:\bar{X}_{1}\to\bar{X}_{2}$ defined by

$x\varphi_{1}\bar{\tau}:=x\tilde{\tau}\varphi_{2}\quad\text{ for all
}x\in\tilde{X}_{1}\,,$

which extends $\tau$. Now, $(\varphi_{1},\varphi_{2})$ is an isomorphism of
pre-actions between $(\tilde{X}_{1},\tilde{X}_{2},\tilde{\tau})$ and
$(\bar{X}_{1},\bar{X}_{2},\bar{\tau})$. ∎

##  6\. High transitivity for amalgams

As in Section 5, we fix an amalgam $\Gamma=\Gamma_{1}*_{\Sigma}\Gamma_{2}$,
and sets of representatives $C_{j}$ of left $\Sigma_{j}$-cosets in
$\Gamma_{j}$ such that $1\in C_{j}$, for $j=1,2$, so that normal forms of
elements of $\Gamma$ are well-defined. We still denote by $N_{C_{j}}$ the set
of elements of $\Gamma$ whose normal form begins with an element of
$C_{j}\setminus\\{1\\}$, for $j=1,2$, so that we have $\Gamma=\Sigma\sqcup
N_{C_{1}}\sqcup N_{C_{2}}$.

Non-degeneracy becomes now essential. Hence, we assume from now on that our
amalgam is non-degenerate.

###  6.1. Using the free globalization towards high transitivity

This section is devoted to a key proposition which will allow us to extend any
given transitive pre-action which is not global to a global one such that the
associated $\Gamma$-action sends one fixed tuple to another fixed tuple.

######  Proposition 6.1.

Suppose $(X_{1},X_{2},\tau)$ is a transitive non-global pre-action, that
$X_{j}$ is a finite union of orbits of a free action
$\bar{X}_{j}\curvearrowleft\Gamma_{j}$, and that the complement
$\bar{X}_{j}-X_{j}$ contains infinitely many $\Gamma_{j}$-orbits. Let
$x_{1},...,x_{k},y_{1},...,y_{k}\in\bar{X}_{1}$ be pairwise distinct points.
Then $(X_{1},X_{2},\tau)$ can be extended to a transitive and global pre-
action $(\bar{X}_{1},\bar{X}_{2},\tilde{\tau})$ so that there is an element
$\gamma\in\Gamma$ such that $x_{i}\gamma^{\pi_{1,\tilde{\tau}}}=y_{i}$, and
the action $\pi_{1,\tilde{\tau}}$ is highly faithful.

Notice that the choice to work in $\bar{X}_{1}$ is arbitrary. We could prove a
similar statement for the $\Gamma_{2}$-action on $\bar{X}_{2}$.

###### Proof.

We will denote the set $\\{x_{1},...,x_{k},y_{1},...,y_{k}\\}$ by $F$. First,
by Theorem 5.19, we find a bijection $\bar{\tau}:\bar{X}_{1}\to\bar{X}_{2}$
such that $(\bar{X}_{1},\bar{X}_{2},\bar{\tau})$ is the free globalization of
$(X_{1},X_{2},\tau)$.

######  Claim.

There exists a path-type element $\gamma$ in $N_{C_{2}}$ such that for every
$x\in F$, the last edge of $\mathsf{path}_{1,x}(\gamma)$ is a treeing edge.

###### Proof of the claim.

Recall the correspondence established in Section 5.3 between path-type
elements and reduced paths of even length. Since
$\mathbf{BS}(\bar{X}_{1},\bar{X}_{2},\bar{\tau})$ is connected and has treeing
edges, it follows from Lemma 2.17 that for every $x\in\bar{X}_{1}$, and every
path-type element $\gamma\in N_{C_{2}}$, there is a path-type extension
$\gamma^{\prime}$ of $\gamma$ such that the last edge of
$\mathsf{path}_{1,x}(\gamma^{\prime})$ is a treeing edge. Now, it suffices to
start with any path-type element $\gamma_{0}\in N_{C_{2}}$, to extend it to a
path-type element $\gamma_{1}$ such that the last edge of
$\mathsf{path}_{1,x_{1}}(\gamma_{1})$ is a treeing edge, then to extend
$\gamma_{1}$ to a path-type element $\gamma_{2}$ such that the last edge of
$\mathsf{path}_{1,y_{1}}(\gamma_{2})$ is a treeing edge, …, and iterate this
extension procedure until we reach an element $\gamma_{2k}\in N_{C_{2}}$ such
that all last edges of $\mathsf{path}_{1,x}(\gamma_{2k})$, for all $x\in F$,
are treeing edges (by Remark 5.11 $(4)$). ∎

Given $x\in\bar{X}_{1}$, and a path-type element $\gamma$ in $N_{C_{2}}$, we
will denote by $\mathcal{H}_{x}(\gamma)$ the half-graph of the last edge of
$\mathsf{path}_{1,x}(\gamma)$.

######  Claim.

There exists a path-type element $\gamma$ in $N_{C_{2}}$ such that for every
$x\in F$, the last edge of $\mathsf{path}_{1,x}(\gamma)$ is a treeing edge,
and the half-trees $\mathcal{H}_{x}(\gamma)$, for $x\in F$, are pairwise
disjoint subgraphs, and all disjoint from $\mathbf{BS}(X_{1},X_{2},\tau)$.

###### Proof of the claim.

We start with a path-type element $\gamma$ in $N_{C_{2}}$ such that for every
$x\in F$, the last edge of $\mathsf{path}_{1,x}(\gamma)$ is a treeing edge.
Since $X_{j}$ is a finite union of $\Gamma_{j}$-orbits for $j=1,2$, the Bass-
Serre graph $\mathbf{BS}(X_{1},X_{2},\tau)$ has finitely many vertices. Hence,
by extending further the path-type element $\gamma$, we can assume that for
every $x\in F$, the half-tree $\mathcal{H}_{x}(\gamma)$ does not intersect
$\mathbf{BS}(X_{1},X_{2},\tau)$.

Notice that, given $x,y\in F$, if the half-trees $\mathcal{H}_{x}(\gamma)$ and
$\mathcal{H}_{y}(\gamma)$ are disjoint, then so are the half-trees
$\mathcal{H}_{x}(\gamma^{\prime})$ and $\mathcal{H}_{y}(\gamma^{\prime})$ for
every path-type extension $\gamma^{\prime}$ of $\gamma$, since
$\mathcal{H}_{x}(\gamma^{\prime})\subseteq\mathcal{H}_{x}(\gamma)$ and
$\mathcal{H}_{y}(\gamma^{\prime})\subseteq\mathcal{H}_{y}(\gamma)$. Hence, it
suffices to prove that, for any $x,y\in F$ with $x\neq y$ and such that
$\mathcal{H}_{x}(\gamma)$ and $\mathcal{H}_{y}(\gamma)$ intersect, there
exists a path-type extension $\gamma^{\prime}$ of $\gamma$ such that
$\mathcal{H}_{x}(\gamma^{\prime})$ and $\mathcal{H}_{y}(\gamma^{\prime})$ are
disjoint. Indeed, an easy induction gives then an extension $\gamma^{(n)}$
such that the half-trees $\mathcal{H}_{x}(\gamma^{(n)})$, for $x\in F$, are
pairwise disjoint.

Take now $x,y\in F$ with $x\neq y$ and such that $\mathcal{H}_{x}(\gamma)$ and
$\mathcal{H}_{y}(\gamma)$ intersect. These half-trees have to be nested.
Indeed, if they are not, $\mathcal{H}_{x}(\gamma)$ contains the antipode of
the last edge of $\mathsf{path}_{y}(\gamma)$, hence contains
$\mathbf{BS}(X_{1},X_{2},\tau)$, which is impossible. Without loss of
generality, we assume
$\mathcal{H}_{x}(\gamma)\subseteq\mathcal{H}_{y}(\gamma)$. We now distinguish
two cases.

  * •

If $\mathcal{H}_{x}(\gamma)\subsetneq\mathcal{H}_{y}(\gamma)$, there is a path
type extension $\gamma^{\prime\prime}$ of $\gamma$ such that
$\mathsf{path}_{1,x}(\gamma)$ and $\mathsf{path}_{1,y}(\gamma^{\prime\prime})$
have the same last edge. We have the product of normal forms

$\gamma^{\prime\prime}=\gamma\cdot(c_{1}\cdots c_{n})\,,$

where $n\geq 2$ is even. Since the amalgam $\Gamma$ is non-degenerate, we can
obtain another normal form $\gamma^{\prime}=\gamma\cdot(c^{\prime}_{1}\cdots
c^{\prime}_{n})$ by replacing a letter $c_{i}$ in the factor $\Gamma_{j}$ such
that $[\Gamma_{j}:\Sigma_{j}]\geq 3$ by another letter $c^{\prime}_{i}$ in
$C_{j}-\\{1\\}$. This change has the effect that
$\mathsf{path}_{1,y}(\gamma^{\prime})$ and
$\mathsf{path}_{1,y}(\gamma^{\prime\prime})$ are distinct reduced paths (which
are both extensions of the $\mathsf{path}_{1,y}(\gamma)$). Hence, since
$\mathcal{H}_{y}(\gamma)$ is a tree, the sub-trees
$\mathcal{H}_{y}(\gamma^{\prime\prime})$ and
$\mathcal{H}_{y}(\gamma^{\prime})$ must be disjoint. Since
$\mathcal{H}_{x}(\gamma)=\mathcal{H}_{y}(\gamma^{\prime\prime})$ we are done.

  * •

If $\mathcal{H}_{x}(\gamma)=\mathcal{H}_{y}(\gamma)$, then
$\mathsf{path}_{1,x}(\gamma)$ and $\mathsf{path}_{1,y}(\gamma)$ have the same
terminal edge, which is

$e:=\overline{x^{\prime}\Sigma_{1}}=\overline{y^{\prime}\Sigma_{1}}\,,\quad\text{
where }\quad x^{\prime}=x\gamma^{\pi_{1,\bar{\tau}}}\text{ and
}y^{\prime}=y\gamma^{\pi_{1,\bar{\tau}}}\,.$

Consequently, one has $y^{\prime}=x^{\prime}\sigma^{\pi_{1,\bar{\tau}}}$ for
some $\sigma\in\Sigma$. Note that, since $x\neq y$, one has $\sigma\neq 1$ and
consider the morphism of pre-actions from the negative translation pre-action
$(\varphi_{1,-},\varphi_{2,-}):(\Gamma_{1}\cup N_{C_{1}},\Gamma_{2}\sqcup
N_{C_{1}},\tau_{-})\to(\bar{X}_{1},\bar{X}_{2},\bar{\tau})$ coming from Remark
5.12, with basepoints $x_{1}=x^{\prime}$ and $x_{2}=x^{\prime}\bar{\tau}$.
Since $e$ is a treeing edge, this morphism is injective by Proposition 5.15.
The half-tree $\mathcal{H}_{x}(\gamma)=\mathcal{H}_{y}(\gamma)$ is thus
isomorphic, via $\mathbf{BS}(\varphi_{1,-},\varphi_{2,-})$, to the half-tree
$\mathcal{H}$ of $\Sigma_{2}$ in the Bass-Serre tree $\mathcal{T}$.

Note that the left translation by $\sigma$ (i.e.
$\gamma^{*}\mapsto\sigma\gamma^{*}$) defines an automorphism of the negative
translation pre-action, which we write as $(\sigma_{1},\sigma_{2})$. The
morphism of graphs $\mathbf{BS}(\sigma_{1},\sigma_{2})$ maps
$\mathsf{path}^{\mathcal{T}}_{2,1_{\Gamma}}(\gamma^{*})$ to
$\mathsf{path}^{\mathcal{T}}_{2,\sigma}(\gamma^{*})$ in $\mathcal{H}$ by Lemma
5.13 (note that these paths both have $\Sigma_{2}$ as first edge). Since the
left $\Gamma$-action on the boundary $\partial\mathcal{T}$ of its Bass-Serre
tree is topologically free, the left $\sigma$ action does not fix the half-
tree $\mathcal{H}$ pointwise. Hence there exists an element
$\gamma^{*}\in\Sigma\sqcup N_{C_{1}}$ such that
$\mathsf{path}^{\mathcal{T}}_{2,1}(\gamma^{*})$ and
$\mathsf{path}^{\mathcal{T}}_{2,\sigma}(\gamma^{*})$ have distinct ranges.
Moreover, up to extending $\gamma^{*}$, we can further assume that
$\gamma_{2}$ is a path type element of $N_{C_{1}}$ so that the element
$\gamma^{\prime}:=\gamma\gamma^{*}$ is a path-type element of $N_{C_{2}}$.

Now, the images of $\mathsf{path}^{\mathcal{T}}_{2,1}(\gamma^{*})$ and
$\mathsf{path}^{\mathcal{T}}_{2,\sigma}(\gamma^{*})$ by
$\mathbf{BS}(\varphi_{1,-},\varphi_{2,-})$ are
$\mathsf{path}_{2,x^{\prime}\bar{\tau}}(\gamma^{*})$ and
$\mathsf{path}_{2,y^{\prime}\bar{\tau}}(\gamma^{*})$ by Lemma 5.13, and these
paths diverge in the half-tree
$\mathcal{H}_{x}(\gamma)=\mathcal{H}_{y}(\gamma)$. Note finally that the
starting edges of these paths are both equal to $e$; this implies that
$\mathsf{path}_{1,x}(\gamma^{\prime})$ and
$\mathsf{path}_{1,y}(\gamma^{\prime})$ don’t have the same range. Hence
$\mathcal{H}_{x}(\gamma^{\prime})$ and $\mathcal{H}_{y}(\gamma^{\prime})$ are
disjoint.

We are done in both cases. ∎

We then modify the bijection $\bar{\tau}$ to get the pre-action
$(\bar{X}_{1},\bar{X}_{2},\tilde{\tau})$ we are looking for. First, given an
element $\gamma$ as in the previous claim, we consider for each $z\in F$ the
morphism of pre-actions from the positive translation pre-action

$(\psi_{1,z},\psi_{2,z}):(\Gamma_{1}\sqcup N_{C_{2}},\Gamma_{2}\cup
N_{C_{2}},\tau_{+})\to(\bar{X}_{1},\bar{X}_{2},\bar{\tau})$

coming from Remark 5.12, with basepoints
$x_{1}=z^{\prime}:=z\gamma^{\pi_{1,\bar{\tau}}}$ and $x_{2}=x_{1}\bar{\tau}$.
Note that the image of this morphism corresponds to the half-graph opposite to
the half-tree $\mathcal{H}_{z}(\gamma)$. Then, we define
$X_{j}^{\prime}=\bigcap_{x\in
F}\operatorname{\mathrm{rng}}(\psi_{j,x})\subset\bar{X}_{j}$, and consider the
restriction $(X_{1}^{\prime},X_{2}^{\prime},\tau^{\prime})$ of
$(\bar{X}_{1},\bar{X}_{2},\bar{\tau})$. Informally speaking, we erase
$\bar{\tau}$ on the $\Sigma_{1}$-orbits corresponding to edges in the half-
trees $\mathcal{H}_{x}(\gamma)$ for $x\in F$. Note that this leaves infinitely
many $\Gamma_{1}$-orbits in $\bar{X}_{1}$ outside
$\operatorname{\mathrm{dom}}(\tau^{\prime})$, respectively infinitely many
$\Gamma_{2}$-orbits in $\bar{X}_{2}$ outside
$\operatorname{\mathrm{rng}}(\tau^{\prime})$, and the pre-action
$(X_{1}^{\prime},X_{2}^{\prime},\tau^{\prime})$ is transitive.

We now extend $\tau^{\prime}$. Pick some free orbits
$z_{1}\Gamma_{2},\ldots,z_{k}\Gamma_{2}$ in $\bar{X}_{2}$ that we add to
$X_{2}^{\prime}$, take $c_{j}$ in $C_{j}-\\{1\\}$ for $j=1,2$, and set
$x_{i}^{\prime}c_{1}\sigma\tau^{\prime}:=z_{i}\vartheta(\sigma)$ and
$y_{i}^{\prime}c_{1}\sigma\tau^{\prime}:=z_{i}c_{2}\vartheta(\sigma)$ for
$i=1,\ldots,k$ and $\sigma\in\Sigma_{1}$. This is possible since the
$\Sigma_{2}$-orbits of the points $z_{i}$ and $z_{i}c_{2}$ are pairwise
disjoint (we use again the freeness of the $\Gamma_{2}$-action), and since the
$\Sigma_{1}$-orbits at $x^{\prime}_{i}c_{1}$, $y^{\prime}_{i}c_{1}$ for $1\leq
i\leq k$ are pairwise disjoint and were not initially in the domain of
$\tau^{\prime}$. Note that the extended version of the pre-action
$(X_{1}^{\prime},X_{2}^{\prime},\tau^{\prime})$ is still transitive. Then we
apply Theorem 5.19 to get an extension
$\tilde{\tau}:\bar{X}_{1}\to\bar{X}_{2}$ of $\tau^{\prime}$ such that
$(\bar{X}_{1},\bar{X}_{2},\tilde{\tau})$ is the free globalization of
$(X_{1}^{\prime},X_{2}^{\prime},\tau^{\prime})$. A computation shows then that
$x_{i}(\gamma c_{1}c_{2}c_{1}^{-1}\gamma^{-1})^{\pi_{1,\tilde{\tau}}}=y_{i}$
for all $i=1,\ldots,k$. Finally, the action $\pi_{1,\tilde{\tau}}$ is highly
faithful by Remark 5.18. ∎

###  6.2. Highly transitive actions of amalgams

From now on, we fix free actions $X_{1}\curvearrowleft^{\pi_{1}}\Gamma_{1}$
and $X_{2}\curvearrowleft^{\pi_{2}}\Gamma_{2}$ with infinitely many orbits. We
endow the set of bijections from $X_{1}$ onto $X_{2}$ with the topology of
pointwise convergence, which is a Polish topology. We then set

$\mathbf{PA}=\\{\tau:X_{1}\to X_{2}\text{
bijective}:\,x\sigma\tau=x\tau\vartheta(\sigma)\text{ for all
}\sigma\in\Sigma_{1}\\}\,.$

In other words, $\mathbf{PA}$ is the set of bijections $\tau:X_{1}\to X_{2}$
such that $(X_{1},X_{2},\tau)$ is a (global) pre-action of $\Gamma$. This is
clearly a closed subset for the topology of pointwise convergence, hence a
Polish space. Recall that every $\tau\in\mathbf{PA}$ induces an action
$X_{j}\curvearrowleft^{\pi_{j,\tau}}\Gamma$ for $j=1,2$. We will focus on the
action $\pi_{1,\tau}$, which we will abbreviate by $\pi_{\tau}$.

######  Definition 6.2.

Let us set

$\displaystyle\mathbf{TA}$
$\displaystyle=\\{\tau\in\mathbf{PA}\colon\pi_{\tau}\text{ is transitive}\\}\
;$ $\displaystyle\mathbf{HFA}$
$\displaystyle=\\{\tau\in\mathbf{PA}\colon\pi_{\tau}\text{ is highly faithful
}\\}\ ;$ $\displaystyle\mathbf{HTA}$
$\displaystyle=\\{\tau\in\mathbf{PA}\colon\pi_{\tau}\text{ is highly
transitive }\\}\ .$

As in the HNN case, the subset $\mathbf{TA}$ isn’t closed for the topology of
pointwise convergence, but we have the following result.

######  Lemma 6.3.

The set $\mathbf{TA}$ is $G_{\delta}$ in $\mathbf{PA}$, hence a Polish space.
Moreover, $\mathbf{TA}\neq\emptyset$.

###### Proof.

Since $X_{1}\curvearrowleft^{\pi_{1}}\Gamma_{1}$ and
$X_{2}\curvearrowleft^{\pi_{2}}\Gamma_{2}$ have infinitely many orbits, there
are $\Gamma_{j}$-equivariant bijections $\varphi_{j}:\Gamma\to X_{j}$ for
$j=1,2$. It then suffices to push-forward the translation pre-action by
$(\varphi_{1},\varphi_{2})$ to get an element of $\mathbf{TA}$ (its Bass-Serre
graph will be isomorphic to the classical Bass-Serre tree and $\pi_{\tau}$
will be conjugated to the translation action $\Gamma\curvearrowleft\Gamma$).
Hence, $\mathbf{TA}$ is non-empty. To show that $\mathbf{TA}$ is $G_{\delta}$
in $\mathbf{PA}$, it suffices to write $\mathbf{TA}=\bigcap_{x,x^{\prime}\in
X_{1}}O_{x,y}$, where for $x,y\in X_{1}$,
$O_{x,y}=\\{\tau\in\mathbf{PA}\,:\,\text{ there exists }\gamma\in\Gamma\text{
such that }x\gamma^{\pi_{\tau}}=y\\}$. Since $O_{x,y}$ is obviously open in
$\mathbf{PA}$ for all $x,y\in X_{1}$, this shows that $\mathbf{TA}$ is a
$G_{\delta}$ subset of $\mathbf{PA}$. ∎

Here comes the theorem proving that our amalgam $\Gamma$ admits a highly
transitive highly faithful action, thus proving Theorem E.

######  Theorem 6.4.

The set $\mathbf{HTA}\cap\mathbf{HFA}$ is dense $G_{\delta}$ in $\mathbf{TA}$.
In particular, $\Gamma$ admits actions which are both highly transitive and
highly faithful.

###### Proof.

For $k\geq 1$ and $x_{1},\dots x_{k},y_{1},\dots,y_{k}\in X_{1}$ pairwise
distinct, the sets,

$V_{x_{1},\dots,x_{k},y_{1},\dots,y_{k}}=\\{\tau\in\mathbf{TA}\,:\,\exists\gamma\in\Gamma\,,\,x_{i}\gamma^{\pi_{\tau}}=y_{i}\text{
for all }1\leq i\leq k\\}$

are obviously open in $\mathbf{TA}$. Similarly, for finite subsets $F$ of
$\Gamma-\\{1\\}$, the sets

$W_{F}=\\{\tau\in\mathbf{TA}\,:\,\exists x\in X_{1}\,,\,xf^{\pi_{\tau}}\neq
x\text{ for all }f\in F\\}$

are also obviously open in $\mathbf{TA}$. Now, using Lemma 2.2, and since
every strongly faithful action of $\Gamma$ is highly faithful by Corollary
2.6, we have

$\mathbf{HTA}\cap\mathbf{HFA}=\underset{\underset{\text{ \ \ \ \ \ \ \ \ \ \ \
\ pairwise distinct}}{F\Subset\Gamma-\\{1\\},\,k\geq 1,\,x_{1},\dots
x_{k},y_{1},\dots,y_{k}\in
X}}{\bigcap}(V_{x_{1},\dots,x_{k},y_{1},\dots,y_{k}}\cap W_{F})\,.$

To conclude, it suffices to show that each set
$(V_{x_{1},\dots,x_{k},y_{1},\dots,y_{k}})\cap\mathbf{HFA}$ is dense in
$\mathbf{TA}$, since this immediately implies that each open set
$(V_{x_{1},\dots,x_{k},y_{1},\dots,y_{k}})\cap W_{F}$ is dense in
$\mathbf{TA}$. To do this, let $\tau\in\mathbf{TA}$ and let $F$ be a finite
subset of $X_{1}$. Fix a finite connected subgraph $\mathcal{G}$ of
$\mathbf{BS}(X_{1},X_{2},\tau)$ containing the edges $z\Sigma_{1}$ for $z\in
F$, and denote by $\tau_{0}$ the restriction of $\tau$ to the union of the
$\Sigma_{1}$-orbits in $X_{1}$ corresponding to the edges of $\mathcal{G}$.
Then apply Proposition 6.1 to the transitive pre-action
$(\operatorname{\mathrm{dom}}(\tau_{0})\cdot\Gamma_{1},\operatorname{\mathrm{rng}}(\tau_{0})\cdot\Gamma_{2},\tau_{0})$,
whose Bass-Serre graph is $\mathcal{G}$, to get an extension $\tau^{\prime}$
such that $\tau^{\prime}\in
V_{x_{1},\dots,x_{k},y_{1},\dots,y_{k}}\cap\mathbf{HFA}$. Moreover, since
$F\subset\operatorname{\mathrm{dom}}(\tau_{0})$, it follows that $\tau$ and
$\tau^{\prime}$ coincide on $F$.∎

######  Remark 6.5.

As in Remark 4.5, one can give a direct proof of the previous theorem without
relying on Baire’s theorem.

##  7\. Highly transitive actions of groups acting on trees

###  7.1. Proofs of Theorem A and B

Let us begin with a few preliminaries.

Suppose we are given an action of a countable group $G$ on a tree
$\mathcal{T}$, and a proper subtree $\mathcal{T}^{\prime}$ such that
$\mathcal{T}^{\prime}$ and $g\mathcal{T}^{\prime}$, where $g\in G$, are either
equal or disjoint subtrees.

We can then can form a “quotient” tree $\bar{\mathcal{T}}$ by shrinking each
subtree $g\mathcal{T}^{\prime}$ to a single vertex, that we will denote by
$(g\mathcal{T}^{\prime})$. The tree $\bar{\mathcal{T}}$ is naturally endowed
with a $G$-action and the “quotient map” $q:\mathcal{T}\to\bar{\mathcal{T}}$
is $G$-equivariant. The image by $q$ of a path in $\mathcal{T}$ is a path
which is obtained by shrinking each subpath contained in a subtree
$g\mathcal{T}^{\prime}$ to the vertex $(g\mathcal{T}^{\prime})$ in
$\bar{\mathcal{T}}$. In case of a geodesic ray, its image by $q$ is either a
geodesic ray in $\bar{\mathcal{T}}$, or a geodesic which ends at a vertex
$(g\mathcal{T}^{\prime})$. Hence $q$ induces a map $\partial
q:\partial\mathcal{T}\to V(\bar{\mathcal{T}})\cup\partial\bar{\mathcal{T}}$.

######  Remark 7.1.

The restriction of $\partial q$ to $(\partial
q)^{-1}(\partial\bar{\mathcal{T}})$ is injective.

###### Proof.

Given $\xi,\xi^{\prime}\in\partial\mathcal{T}$ such that $\partial q(\xi)$ and
$\partial q(\xi^{\prime})$ lie in $\partial\bar{\mathcal{T}}$, consider
geodesic rays $\omega,\omega^{\prime}$ in $\mathcal{T}$ tending to
$\xi,\xi^{\prime}$. Each ray contains all edges of its image under $q$. Hence
$\partial q(\xi)=\partial q(\xi^{\prime})$ implies that $\omega$ and
$\omega^{\prime}$ have infinitely many common edges, and therefore
$\xi=\xi^{\prime}$. ∎

One can also notice, although we do not need this fact below, that $\partial
q$ is continuous at each point $\xi\in(\partial
q)^{-1}(\partial\bar{\mathcal{T}})$. In case $\mathcal{T}^{\prime}$ is
bounded, one has in fact $\partial
q:\partial\mathcal{T}\to\partial\bar{\mathcal{T}}$, and $\partial q$ is
continuous and injective.

######  Lemma 7.2.

In the context above, assume that $G\curvearrowright\mathcal{T}$ is a minimal
action. Then:

  1. (1)

if $G\curvearrowright\mathcal{T}$ is of general type, then so is
$G\curvearrowright\bar{\mathcal{T}}$;

  2. (2)

if $G\curvearrowright\partial\mathcal{T}$ is topologically free, then so is
$G\curvearrowright\partial\bar{\mathcal{T}}$.

###### Proof.

Assume first $G\curvearrowright\mathcal{T}$ is of general type, in order to
prove (1). The hypotheses on $\mathcal{T}^{\prime}$ guarantee the existence of
an edge $e$ in $\mathcal{T}$ which lies outside all translates
$g\mathcal{T}^{\prime}$. By minimality of $G\curvearrowright\mathcal{T}$,
there exists a hyperbolic element $h\in G$ whose axis in $\mathcal{T}$
contains $e$. Pick $g_{1},g_{2}\in G$ which induce transverse hyperbolic
automorphisms of $\mathcal{T}$. For $n$ sufficiently large,
$h_{1}=g_{1}^{n}hg_{1}^{-n}$ and $h_{2}=g_{2}^{n}hg_{2}^{-n}$ induce
transverse hyperbolic automorphisms of $\mathcal{T}$. Moreover, their axes do
contain edges in the orbit of $e$, so that their images by $q$ lie in
$\partial\bar{\mathcal{T}}$. Hence, by Remark 7.1, $h_{1}$ and $h_{2}$ induce
transverse hyperbolic automorphisms of $\bar{\mathcal{T}}$. This proves that
$G\curvearrowright\bar{\mathcal{T}}$ is of general type.

Assume now that $G\curvearrowright\partial\mathcal{T}$ topologically free, in
order to prove (2). Then assume that $g\in G$ fixes a half-tree
$\mathcal{H}^{\prime}$ in $\bar{\mathcal{T}}$, corresponding to some edge $e$
in $\mathcal{T}^{\prime}$, pointwise. Notice that $e$ is an edge of
$\mathcal{T}$ that $q$ does not shrink, and denote by $\mathcal{H}$ its half-
tree in $\mathcal{T}$. One has $q(\mathcal{H})=\mathcal{H}^{\prime}$. By
minimality of $G\curvearrowright\mathcal{T}$, the edges of the orbit $G\cdot
e$ which lie in $\mathcal{H}$ do generate $\mathcal{H}$. Since they also lie
in $\mathcal{H}^{\prime}$, they are fixed pointwise by $g$, therefore the
half-tree $\mathcal{H}$ itself is fixed pointwise by $g$. Since
$G\curvearrowright\partial\mathcal{T}$ is topologically free, $g$ has to be
the trivial element, and this proves that
$G\curvearrowright\partial\bar{\mathcal{T}}$ is topologically free. ∎

We now recall the statement of Theorem A before proving it.

######  Theorem.

Let $\Gamma\curvearrowright\mathcal{T}$ be a minimal action of general type of
a countable group $\Gamma$ on a tree $\mathcal{T}$. If the action on the
boundary $\Gamma\curvearrowright\partial\mathcal{T}$ is topologically free,
then $\Gamma$ admits a highly transitive and highly faithful action; in
particular, $\Gamma$ is highly transitive.

###### Proof of Theorem A.

Let us consider an edge $e$ in $\mathcal{T}$. The complement of the orbits
$\Gamma\cdot e$ and $\Gamma\cdot\bar{e}$ in $E(\mathcal{T})$ is either empty,
or generates a disjoint union of subtrees of $\mathcal{T}$. Since each of
these subtrees contains some endpoint of $e$, there are at most two orbits of
subtrees. Hence by applying Lemma 7.2 zero, one, or two times, one gets a tree
$\bar{\mathcal{T}}$ endowed with a $\Gamma$-action which is still of general
type, and such that $\Gamma\curvearrowright\partial\bar{\mathcal{T}}$ is
topologically free. Moreover, the action on $E(\bar{\mathcal{T}})$ is
transitive. Now, the quotient $\Gamma\backslash\bar{\mathcal{T}}$ is either a
segment or a loop, the fundamental group of the corresponding graph of groups,
which is either an amalgam or an HNN extension, is isomorphic to $\Gamma$, and
$\mathcal{T}$ is the associated Bass-Serre tree. Applying Theorem E or Theorem
C, we finally get that $\Gamma$ admits a highly transitive and highly faithful
action. ∎

We can now also prove Theorem B, but let us first recall its statement.

######  Theorem.

Let $\Gamma\curvearrowright\mathcal{T}$ be a faithful minimal action of
general type of a countable group $\Gamma$ on a tree $\mathcal{T}$. The
following are equivalent

  1. (1)

$\mathrm{td}(\Gamma)\geq 4$;

  2. (2)

$\Gamma$ is highly transitive;

  3. (3)

$\Gamma$ is MIF;

  4. (4)

$\Gamma\curvearrowright\partial\mathcal{T}$ is topologically free.

###### Proof of Theorem B.

The implication (2) $\implies$ (1) is clear. The implication (1) $\implies$
(4) is Le Boudec and Matte Bon’s main result [LBMB19, Thm 1.4]. The
implication (4) $\implies$ (2) is a consequence of Theorem A. So (1), (2) and
(4) are all equivalent.

To prove that these three statements are also equivalent to (3), note that
Theorem A shows moreover that, under the assumption (4), the group $\Gamma$
admits a highly transitive _highly faithful_ action. So for such an action,
all its elements have infinite support, which by [HO16, Corollary 5.8] implies
that $\Gamma$ is MIF. The implication (4) $\implies$ (3) thus holds. Finally,
the implication (3) $\implies$ (4) follows from [LBMB19, Proposition 3.7] ∎

###  7.2. Corollary F and its implication of former results

We now turn to the proof of a lemma which directly implies Corollary F via
Theorem A, and then we check that Corollary F applies to all groups acting on
trees which can be proven to be highly transitive by previous results quoted
in the introduction.

Recall that, given a subtree $\mathcal{U}$ of $\mathcal{T}$, we denote by
$G_{\mathcal{U}}$ the pointwise stabilizer of $\mathcal{U}$ in $G$. The
following lemma is a generalization of Prop. 19 (iv) and Prop. 20 (iv) from
[HP11].

######  Lemma 7.3.

Let $G\curvearrowright\mathcal{T}$ be a faithful and minimal action such that
$G$ contains a hyperbolic element $h$. If there exist a bounded subtree
$\mathcal{B}$ and a vertex $u$ in $\mathcal{B}$ such that $G_{\mathcal{B}}$ is
core-free in $G_{u}$, then the induced action
$G\curvearrowright\partial\mathcal{T}$ is topologically free.

###### Proof.

Let $\mathcal{B}^{\prime}$ be the union of the translates $g\mathcal{B}$ for
$g\in G_{u}$. This is a subtree, since all $g\mathcal{B}$ contain $u$, which
is $G_{u}$-invariant and contained in the ball of radius
$\operatorname{\mathrm{diam}}(\mathcal{B})$ centered at $u$. Let $g_{0}$ be an
element of $G$ fixing a half-tree $\mathcal{H}$ pointwise. Up to conjugating
by a suitable power of $h$, we may and will assume that $\mathcal{H}$ contains
$\mathcal{B}^{\prime}$, so that $g_{0}$ is in $G_{\mathcal{B}^{\prime}}$. Now,
as $G_{\mathcal{B}}$ is core-free in $G_{u}$, we have
$G_{\mathcal{B}^{\prime}}=\bigcap_{g\in G_{u}}G_{g\mathcal{B}}=\bigcap_{g\in
G_{u}}gG_{\mathcal{B}}g^{-1}=\\{1\\}$. Thus, we get $g_{0}=1$, which proves
that $G\curvearrowright\partial\mathcal{T}$ is topologically free by Corollary
2.14. ∎

We now prove that Corollary F applies to all groups acting on trees which are
highly transitive via the combination of the results of Minasyan-Osin [MO15]
and Hull-Osin [HO16].

######  Proposition 7.4.

Let $\Gamma$ be a countable group acting minimally on a tree $\mathcal{T}$.
Suppose that

  1. (i)

$\Gamma$ is not virtually cyclic,

  2. (ii)

$\Gamma$ does not fix any point of $\partial\mathcal{T}$,

  3. (iii)

there exist vertices $u,v$ of $\mathcal{T}$ such that the stabilizer
$\Gamma_{[u,v]}$ is finite;

  4. (iv)

the finite radical of $\Gamma$ is trivial.

Then the action $\Gamma\curvearrowright\mathcal{T}$ is faithful, of general
type, and there exists a bounded subtree $\mathcal{B}$ such that
$\Gamma_{\mathcal{B}}$ is trivial. In particular, $\Gamma_{\mathcal{B}}$ is
core-free in $\Gamma_{u}$ for every vertex $u$ in $\mathcal{B}$.

###### Proof.

The action $\Gamma\curvearrowright\mathcal{T}$ cannot be elliptic. Indeed, if
it would, then $\mathcal{T}$ would be a singleton, $\Gamma$ would be finite by
(iii), and this contradicts (i). Furthermore, this action cannot be lineal,
because of (i) and minimality, nor parabolic, nor quasi-parabolic, because of
(ii). Hence, $\Gamma\curvearrowright\mathcal{T}$ is of general type.

Starting with a finite stabilizer $\Gamma_{[u,v]}$ given by (iii), one
observes that $\bigcap_{\gamma\in\Gamma}\gamma\Gamma_{[u,v]}\gamma^{-1}$ is
contained in the finite radical, hence trivial by (iv). In particular,
$\Gamma\curvearrowright\mathcal{T}$ is faithful. Then, there is a finite
subset $F\Subset\Gamma$, containing $1$, such that $\bigcap_{f\in
F}f\Gamma_{[u,v]}f^{-1}$ is already trivial, hence $\bigcap_{f\in
F}\Gamma_{f\cdot[u,v]}$ is trivial. Now, we are done by considering the
smallest subtree $\mathcal{B}$ of $\mathcal{T}$ containing the geodesics
$f\cdot[u,v]$, for $f\in F$. ∎

Second, we prove that Corollary F applies to all groups which are highly
transitive thanks to [FMS15]. This is a straightforward consequence of the
following result.

######  Proposition 7.5.

Let a countable group $\Gamma$ act without inversion on a tree $\mathcal{T}$,
and consider a set $R\subset E(\mathcal{T})$ of representatives of the edges
of the quotient graph $\Gamma\backslash\mathcal{T}$. Assume that $\Gamma_{v}$
is infinite and $\Gamma_{e}$ is highly core-free in $\Gamma_{v}$, for every
couple $(e,v)$ where $e\in R$ and $v$ is one of its endpoints. Then, there
exists a subtree $\mathcal{T}^{\prime}$ of $\mathcal{T}$ such that:

  1. (1)

the action $\Gamma\curvearrowright\mathcal{T}^{\prime}$ is faithful, of
general type, and minimal;

  2. (2)

there exist a bounded subtree $\mathcal{B}$ of $\mathcal{T}^{\prime}$ and
$u\in V(\mathcal{B})$ such that $\Gamma_{\mathcal{B}}$ is core-free in
$\Gamma_{u}$.

###### Proof.

First, assume that $\gamma\in\Gamma$ fixes $\mathcal{T}$ pointwise. Then, one
has $\gamma\in\Gamma_{v}$ for some endpoint $v$ of an edge $e\in R$. As
$\Gamma_{e}$ is highly core-free in $\Gamma_{v}$, the $\Gamma_{v}$-action on
the orbit $\Gamma_{v}\cdot e$, which is conjugate to
$\Gamma_{v}\curvearrowright\Gamma_{v}/\Gamma_{e}$, is highly faithful. Hence,
we get $\gamma=1$, so that $\Gamma\curvearrowright\mathcal{T}$ is faithful.

Second, let us consider any edge $e\in R$, and set $v=s(e)$, $w=r(e)$. By high
core-freeness, the indexes $[\Gamma_{v}:\Gamma_{e}]$ and
$[\Gamma_{w}:\Gamma_{e}]$ are both infinite. Thus, there exist elements
$g_{1},g_{2}\in\Gamma_{w}$ and $h_{1},h_{2}\in\Gamma_{v}$ such that the edges
$e,g_{1}^{-1}e,g_{2}^{-1}e,h_{1}e,h_{2}e$ are pairwise distinct. Notice that
$(g_{1}^{-1}e,\bar{e},h_{1}e)$ and $(g_{2}^{-1}e,\bar{e},h_{2}e)$ are oriented
paths. For $j=1,2$, the element $h_{j}g_{j}$ is hyperbolic and its axis
contains $(g_{j}^{-1}e,\bar{e},h_{j}e)$. We have got transverse hyperbolic
elements $h_{1}g_{1},h_{2}g_{2}$, which proves that
$\Gamma\curvearrowright\mathcal{T}$ is of general type.

In fact, $\Gamma_{v}$ is infinite and $\Gamma_{e}$ is highly core-free in
$\Gamma_{v}$, for every couple $(e,v)$ where $e$ is any edge of $\mathcal{T}$
and $v$ is one of its endpoints. Notice this property passes to the smallest
subtree $\mathcal{T}^{\prime}$ of $\mathcal{T}$ containing the axes of all
hyperbolic elements in $\Gamma$. This subtree is $\Gamma$-invariant, and the
action $\Gamma\curvearrowright\mathcal{T}^{\prime}$ is still faithful and of
general type. Of course, it is also minimal, and Assertion (2) is trivially
satisfied when $\mathcal{B}$ is any segment (that is, any subtree with exactly
two vertices). ∎

Finally, we state two natural consequences of Corollary F when considering the
natural actions of HNN extensions (resp. amalgams) on their Bass-Serre tree.

######  Corollary 7.6.

Consider a non-ascending HNN extension
$\Gamma=\operatorname{HNN}(H,\Sigma,\vartheta)$. If one of the subgroups
$\Sigma,\vartheta(\Sigma)$ is core-free in $H$, then $\Gamma$ admits a highly
transitive and highly faithful action; in particular, $\Gamma$ is highly
transitive.

###### Proof.

Apply Corollary F to the tree induced by the edge $\Sigma$ and the vertex $H$,
or to the tree induced by the edge $\vartheta(\Sigma)$ and the vertex $H$. ∎

######  Corollary 7.7.

Consider a non-degenerate amalgam $\Gamma=\Gamma_{1}*_{\Sigma}\Gamma_{2}$. If
$\Sigma$ is core-free in one factor $\Gamma_{j}$, then then $\Gamma$ admits a
highly transitive and highly faithful action; in particular, $\Gamma$ is
highly transitive.

###### Proof.

Apply Corollary F to the tree induced by the edge $\Sigma$ and the vertex
$\Gamma_{j}$. ∎

##  8\. Examples and applications

As mentioned in the Introduction, it is worth giving examples of groups which
are highly transitive thanks to Theorem A, or to its consequences, but for
which previous results from [MO15, HO16, FMS15, GGS20] do not apply. In
particular, we will prove that some groups are neither acylindrically
hyperbolic nor linear. To that end, we will use the following well-known
results.

######  Proposition 8.1.

[Osi16, Corollary 1.5] If a group $\Gamma$ is acylindrically hyperbolic, then
so is any s-normal subgroup of $\Gamma$.

Let us recall that a subgroup $\Lambda\leq\Gamma$ is called s-normal if for
every $\gamma\in\Gamma$, the subgroup $\gamma\Lambda\gamma^{-1}\cap\Lambda$ is
infinite. Every infinite normal subgroup is clearly s-normal.

######  Proposition 8.2.

Every acylindrically hyperbolic group contains a non-abelian free subgroup. In
particular, every acylindrically hyperbolic group is non-amenable.

The latter proposition can either be proved by a standard ping-pong argument,
or deduced from Theorem 6.8 and Theorem 8.1 in [DGO17], which imply that every
acylindrically hyperbolic group is SQ-universal (see also the discussion
around Conjecture 9.6 in the same book).

Let us recall that a group is called linear over a field $k$ if it is
isomorphic to a subgroup of ${\rm GL}(V)$, where $V$ is a finite dimensional
$k$-vector space. Note that if $k^{\prime}$ is an extension of $k$ then any
group linear over $k$ is linear over $k^{\prime}$. Hence, if a group is linear
over $k$ then it is also linear over the algebraic closure of $k$.

A group $\Gamma$ is called linear if there exists a field $k$ such that
$\Gamma$ is linear over $k$. It follows from the preceding discussion that
$\Gamma$ is linear if and only if there exists an algebraically closed field
$k$ such that $\Gamma$ is linear over $k$.

###  8.1. Examples around Baumslag-Solitar groups

####  8.1.1. Baumslag-Solitar groups themselves

Let us recall the definition: for any $m,n\in\mathbb{Z}^{*}$, the Baumslag-
Solitar group with parameters $m,n$ is

${\rm BS}(m,n):=\langle a,b\,|\,ab^{m}a^{-1}=b^{n}\rangle.$

Hull and Osin asked what the transitivity degree of Baumslag-Solitar groups is
[HO16, Question 6.3], and noted that it was actually unknown whether
${\rm{BS(2,3)}}$ is highly transitive or not. We completely answer this
question in Proposition 8.8 and Corollary 8.12 below.

Notice that ${\rm BS}(m,n)$ is isomorphic to
$\operatorname{HNN}(\mathbb{Z},n\mathbb{Z},\vartheta)$, where
$\vartheta(nq)=mq$ for all $q\in\mathbb{Z}$, and the isomorphism from
$\operatorname{HNN}(\mathbb{Z},n\mathbb{Z},\vartheta)$ to ${\rm BS}(m,n)$ is
given by $t\mapsto a$ and $q\mapsto b^{q}$ for $q\in\mathbb{Z}$. We will
freely identify ${\rm BS}(m,n)$ to this HNN extension below without recalling
it explicitly. Hence, ${\rm BS}(m,n)$ has a natural action on the Bass-Serre
tree of this HNN extension, which we denote by $\mathcal{T}_{m,n}$.

######  Remark 8.3.

The following facts are well-known:

  * •

${\rm BS}(m,n)$ is solvable if and only if $|m|=1$ or $|n|=1$;

  * •

${\rm BS}(m,n)$ is icc if and only if $|m|\neq|n|$;

  * •

${\rm BS}(m,n)$ is residually finite if and only if $|m|=1$, $|n|=1$, or
$|m|=|n|$; see [Mes72];

  * •

${\rm BS}(m,n)$ is non-linear whenever $|m|\neq 1$, $|n|\neq 1$, and
$|m|\neq|n|$; this is a consequence of the former fact and Malcev’s theorem
[Mal40].

######  Remark 8.4.

Hull and Osin observed that ${\rm BS}(m,n)$ is never acylindrically hyperbolic
(for $m,n\in\mathbb{Z}^{*}$); this is [Osi16, Example 7.4]. Let us recall the
argument: since the cyclic subgroup $\langle b\rangle$ is s-normal, the group
${\rm BS}(m,n)$ is not acylindrically hyperbolic by Proposition 8.1.

Let us note the following result for later use.

######  Lemma 8.5.

For all $m,n\in\mathbb{Z}^{*}$ and $r\geq 1$, the subgroup $\langle
b^{r}\rangle$ is s-normal in ${\rm BS}(m,n)$.

###### Proof.

Let $\gamma$ be any element of ${\rm BS}(m,n)$, with normal form
$b^{s_{0}}a^{\varepsilon_{1}}b^{s_{1}}\cdots a^{\varepsilon_{k}}b^{s_{k}}$. It
is easy to check that $\gamma b^{rm^{k}n^{k}}\gamma^{-1}$ is still a non-
trivial power of $b$, say $b^{s}$. Then, $\langle
b^{r}\rangle\cap\gamma\langle b^{r}\rangle\gamma^{-1}$ contains $\langle
b^{rm^{k}n^{k}s}\rangle$, hence is infinite. ∎

Let us now turn to a crucial lemma before stating our result. It is due to de
la Harpe and Préaux [HP11, Lem. 21], but we include a proof for the reader’s
convenience.

######  Lemma 8.6 (de la Harpe-Préaux).

The action ${\rm BS}(m,n)\curvearrowright\partial\mathcal{T}_{m,n}$ is
topologically free if and only if $|n|\neq|m|$.

###### Proof.

($\Longleftarrow$) Since $|n|\neq|m|$, either $n\nmid m$ or $m\nmid n$. Assume
that $n\nmid m$ and let $d=\gcd(n,m)$ and $n=dn_{0}$, $m=dm_{0}$. Since
$n\nmid m$ we have $|n_{0}|\geq 2$. In particular, we have $\Sigma\neq H$,
hence there are several positive edges in the star at any vertex in
$\mathcal{T}_{m,n}$. Consequently every half-tree in $\mathcal{T}_{m,n}$
contains a a half-tree corresponding to a positive edge. Now, let
$\gamma\in{\rm BS}(m,n)$, suppose $\gamma$ fixes pointwise a half-tree. Since
the action of ${\rm BS}(m,n)$ is transitive on the positive edges of
$\mathcal{T}_{m,n}$, we may assume that the fixed half-tree $\mathcal{H}$
contains the one given by the edge $\Sigma$. For all $k\geq 1$, one has
$\mathsf{path}_{1}(t^{k})\subset\mathcal{H}$, hence $\gamma$ fixes
$\mathsf{path}_{1}(t^{k})$ pointwise. It follows that $\gamma\in\Sigma\cap
t^{k}\Sigma t^{-k}=n_{0}m_{0}^{k}d\mathbb{Z}$, for all $k\geq 1$. Hence,
$\gamma=1$, since $|n_{0}|\geq 2$. In the case $m\nmid n$ the proof is similar
(we could also deduce this case from the isomorphism ${\rm BS}(m,n)\simeq{\rm
BS}(n,m)$).

($\Longrightarrow$) Suppose that $|n|=|m|$, so that $\Sigma=\vartheta(\Sigma)$
and $\vartheta=\pm\mathord{\operatorname{id}}$. In this case, $\Sigma$ is a
non-trivial normal subgroup of ${\rm BS}(m,n)$, and $\mathcal{T}_{m,n}$ itself
is fixed pointwise by any element of $\Sigma$. ∎

######  Remark 8.7.

The lemma immediately implies that $\langle b\rangle$ is a core-free subgroup
of ${\rm BS}(m,n)$ whenever $|n|\neq|m|$. Indeed, the action on ${\rm
BS}(m,n)\curvearrowright\partial\mathcal{T}_{m,n}$ being topologically free,
the action ${\rm BS}(m,n)\curvearrowright\mathcal{T}_{m,n}$ is faithful. Hence
$\langle b\rangle$ is core-free in ${\rm BS}(m,n)$, since the conjugates of
$\langle b\rangle$ in ${\rm BS}(m,n)$ are exactly the vertex stabilizers. See
also Lemma 8.13.

Our first new examples of highly transitive groups are given by the following
result.

######  Proposition 8.8.

Let $m,n\in\mathbb{Z}^{*}$. The following are equivalent:

  1. (i)

$|m|\neq 1$, $|n|\neq 1$, and $|m|\neq|n|$;

  2. (ii)

${\rm BS}(m,n)$ admits a highly transitive and highly faithful action;

  3. (iii)

${\rm BS}(m,n)$ is highly transitive;

  4. (iv)

${\rm BS}(m,n)$ is non-solvable and icc.

###### Proof.

The implication (i)$\implies$(ii) is a direct consequence of Theorem C and
Lemma 8.6. Then, (ii)$\implies$(iii) is trivial, (iii)$\implies$(iv) results
from classical obstructions to high faithfulness recalled in the Introduction,
and (iv)$\implies$(i) results from Remark 8.3. ∎

######  Remark 8.9.

As reminded in Remark 8.3 and Remark 8.4, the highly transitive groups arising
in Proposition 8.8 are non-acylindrically hyperbolic and non-linear. Moreover,
edge-stabilizers are not highly core-free in their endpoints stabilizers (they
have finite index). Hence, results from [MO15, HO16, FMS15, GGS20] do not
apply.

######  Remark 8.10.

The following lemma proves that Corollary F cannot apply to the action ${\rm
BS}(m,n)\curvearrowright\mathcal{T}_{m,n}$. Consequently, Theorem A is
stronger than Corollary F. Note Baumslag-Solitar groups are our only examples
which testify this fact.

######  Lemma 8.11.

Set $\Gamma:={\rm BS}(m,n)$. If $\mathcal{B}$ is any bounded subtree of
$\mathcal{T}_{m,n}$ and $u$ is any vertex of $\mathcal{B}$, then the pointwise
stabilizer $\Gamma_{\mathcal{B}}$ is not core-free in $\Gamma_{u}$.

###### Proof.

There exists a positive integer $r$ such that $\mathcal{B}$ is contained in
the ball $\mathcal{B}(r)$ of radius $r$ at $\langle b\rangle$. Then, every
stabilizer $\Gamma_{v}$, where $v\in V(\mathcal{B}(r))$, is a conjugate
subgroup $\gamma\langle b\rangle\gamma^{-1}$, where the normal form of
$\gamma$ contains at most $r$ occurrences of $a^{\pm 1}$. Consequently
$\gamma^{-1}b^{m^{r}n^{r}}\gamma$ is still a power of $b$, so that
$b^{m^{r}n^{r}}$ lies in $\gamma\langle b\rangle\gamma^{-1}=\Gamma_{v}$. This
proves that $b^{m^{r}n^{r}}$ lies in the pointwise stabilizer
$\Gamma_{\mathcal{B}(r)}$. Now, for every $\gamma\in\Gamma_{u}$, one has
$\gamma\Gamma_{\mathcal{B}}\gamma^{-1}=\Gamma_{\gamma\mathcal{B}}\supseteq\Gamma_{\mathcal{B}(r)}$,
since $\gamma\mathcal{B}\subseteq\mathcal{B}(r)$. Thus all conjugates
$\gamma\Gamma_{\mathcal{B}}\gamma^{-1}$ where $\gamma\in\Gamma_{u}$ contain
$b^{m^{r}n^{r}}$, so $\Gamma_{\mathcal{B}}$ is not core-free in $\Gamma_{u}$.
∎

Let us now complete the answer to Hull and Osin’s question. As they noticed in
[HO16, Lemma 4.2(b) and Corollary 4.6], infinite non-icc groups and infinite
residually finite solvable groups have transitivity degree $1$. Hence we can
compute the transitivity degree of all Baumslag-Solitar groups.

######  Corollary 8.12.

Let $m,n\in\mathbb{Z}^{*}$. The following hold.

  1. (1)

If $|n|=1$ or $|m|=1$ or $|n|=|m|$, then ${\rm td}({\rm BS}(m,n))=1$.

  2. (2)

In the other cases, ${\rm BS}(m,n)$ is highly transitive, so ${\rm td}({\rm
BS}(m,n))=+\infty$.

###### Proof.

(1) If $|n|=1$ or $|m|=1$, the group ${\rm BS}(m,n)$ is infinite, residually
finite, and solvable, hence ${\rm td}({\rm BS}(m,n))=1$. If $|n|=|m|$, the
group ${\rm BS}(m,n)$ is infinite and non-icc, hence ${\rm td}({\rm
BS}(m,n))=1$.

(2) This follows from Proposition 8.8 directly. ∎

####  8.1.2. Amalgams with Baumslag-Solitar groups

Let us now turn to examples of highly transitive groups given by amalgams. Let
us begin by some more preliminaries.

######  Lemma 8.13.

Let $m,n\in\mathbb{Z}^{*}$. If $|n|\neq|m|$, the subgroup $\langle b\rangle$
is highly core-free in ${\rm BS}(m,n)$.

Notice this is essentially the same example as the one given in [HO16,
Corollary 5.12]. As the action on ${\rm
BS}(m,n)\curvearrowright\partial\mathcal{T}_{m,n}$ is topologically free, it
is a particular case of a general phenomenon described in the following lemma.

######  Lemma 8.14.

Let $\Gamma\curvearrowright\mathcal{T}$ be a minimal action of a countable
group $\Gamma$ on a tree $\mathcal{T}$. If the action on the boundary
$\Gamma\curvearrowright\partial\mathcal{T}$ is topologically free, then for
every vertex $v$ in $\mathcal{T}$, the stabilizer $\Gamma_{v}$ is highly core-
free in $\Gamma$.

###### Proof.

We have to prove that the action $\Gamma\curvearrowright\Gamma/\Gamma_{v}$ is
highly faithful; by Corollary 2.6 it is sufficient to prove it is strongly
faithful. Notice the orbit $\Gamma v$ in $\mathcal{T}$ meets every half-tree
in $\mathcal{T}$ by minimality of the action
$\Gamma\curvearrowright\mathcal{T}$, and that $\Gamma\curvearrowright\Gamma v$
is conjugate to $\Gamma\curvearrowright\Gamma/\Gamma_{v}$.

Take any non-trivial elements $\gamma_{1},\ldots,\gamma_{k}\in\Gamma$. Let us
start with any half-tree $\mathcal{H}_{0}$. Then, since the fixed points of
$\gamma_{1}$ form a subtree, and since
$\Gamma\curvearrowright\partial\mathcal{T}$ is topologically free, there
exists a half-tree $\mathcal{H}_{1}\subseteq\mathcal{H}_{0}$, all of whose
vertices are moved by $\gamma_{1}$. Then applying the same argument to
$\gamma_{2}$, we get a half-tree $\mathcal{H}_{2}\subseteq\mathcal{H}_{1}$,
all of whose vertices are moved by $\gamma_{1}$ and $\gamma_{2}$. And so on,
and so forth, we finish with a half-tree $\mathcal{H}_{k}$, all of whose
vertices are moved by all elements $\gamma_{1},\ldots,\gamma_{k}$. Finally
$\mathcal{H}_{k}$ contains a point of $\Gamma v$, which is moved by all
elements $\gamma_{1},\ldots,\gamma_{k}$. This proves that
$\Gamma\curvearrowright\Gamma v$ is strongly faithful. ∎

We also need a general fact about s-normality in amalgams.

######  Lemma 8.15.

Let us consider an amalgam $\Gamma=\Gamma_{1}*_{\Sigma}\Gamma_{2}$. If all
infinite subgroups $\Sigma^{\prime}<\Sigma$ are s-normal in both $\Gamma_{1}$
and $\Gamma_{2}$, then they are also all s-normal in $\Gamma$.

###### Proof.

Let $\Sigma_{0}$ be any infinite subgroup of $\Sigma$, and let $\gamma$ be any
element of $\Gamma$, that we write as a product
$\gamma=\gamma_{1}\cdots\gamma_{n}$ of elements of $\Gamma_{1}$ or
$\Gamma_{2}$. Set
$\Sigma_{k}=\Sigma_{k-1}\cap\gamma_{k}^{-1}\Sigma_{k-1}\gamma_{k}$ for
$k=1,\ldots,n$. Let us prove by induction that $\Sigma_{k}$ is infinite, and
contained in
$\Sigma_{0}\cap(\gamma_{1}\cdots\gamma_{k})^{-1}\Sigma_{0}(\gamma_{1}\cdots\gamma_{k})$
for $k=0,\ldots,n$.

For $k=0$, the group $\Sigma_{0}$ has been supposed infinite, and it coincides
with the intersection
$\Sigma_{0}\cap(\gamma_{1}\cdots\gamma_{k})^{-1}\Sigma_{0}(\gamma_{1}\cdots\gamma_{k})$
in this case. Then, for $k\geq 1$, the subgroup $\Sigma_{k-1}$ is infinite by
induction hypothesis, therefore $\Sigma_{k-1}$ is s-normal in $\Gamma_{1}$ and
$\Gamma_{2}$. Consequently, the subgroup
$\Sigma_{k}=\Sigma_{k-1}\cap\gamma_{k}^{-1}\Sigma_{k-1}\gamma_{k}$ is
infinite. Moreover, one has

$\displaystyle\Sigma_{k}$
$\displaystyle=\Sigma_{k-1}\cap\gamma_{k}^{-1}\Sigma_{k-1}\gamma_{k}$
$\displaystyle\subseteq\Sigma_{0}\cap(\gamma_{1}\cdots\gamma_{k-1})^{-1}\Sigma_{0}(\gamma_{1}\cdots\gamma_{k-1})\cap\gamma_{k}^{-1}\Sigma_{0}\gamma_{k}\cap(\gamma_{1}\cdots\gamma_{k})^{-1}\Sigma_{0}(\gamma_{1}\cdots\gamma_{k})$

by induction hypothesis, whence
$\Sigma_{k}\subseteq\Sigma_{0}\cap(\gamma_{1}\cdots\gamma_{k})^{-1}\Sigma_{0}(\gamma_{1}\cdots\gamma_{k})$.

Finally, applying the result with $k=n$, we get that $\Sigma_{n}$ is infinite
and $\Sigma_{n}\subseteq\Sigma_{0}\cap\gamma^{-1}\Sigma_{0}\gamma$, which
proves that $\Sigma_{0}$ is s-normal in $\Gamma$, as desired. ∎

Let us finally turn to our examples of highly transitive amalgams.

######  Proposition 8.16.

Let $m,n,k\in\mathbb{Z}^{*}$, and let $\Lambda$ be a countable group
containing a proper infinite cyclic subgroup $\langle c\rangle$. The amalgam
$\Gamma={\rm BS}(m,n)*_{\langle b^{k}=c\rangle}G$ has the following
properties:

  1. (1)

if $|n|\neq|m|$, then $\Gamma$ admits an action which is both highly
transitive and highly faithful;

  2. (2)

if $|n|\neq|m|$ and $\langle c\rangle$ is s-normal in $\Lambda$, then $\Gamma$
is not acylindrically hyperbolic;

  3. (3)

if $|m|\neq 1$, $|n|\neq 1$, and $|m|\neq|n|$, then $\Gamma$ is not a linear
group.

###### Proof.

(1) The amalgam $\Gamma$ is non-degenerate. By Lemma 8.13, or Remark 8.7,
$\langle b^{k}\rangle$ is a core-free subgroup of ${\rm BS}(m,n)$. Then
Corollary F, or in this case Corollary 7.7, implies that $\Gamma$ admits an
action which is both highly transitive and highly faithful.

(2) By Lemma 8.5, every non-trivial subgroup of $\langle b\rangle$ is s-normal
in ${\rm BS}(m,n)$. Furthermore, in $\Lambda$, for any $\lambda\in\Lambda$,
the intersection $\langle c\rangle\cap\lambda\langle c\rangle\lambda^{-1}$ is
infinite cyclic, say generated by $c_{\lambda}$, since $\langle c\rangle$ is
s-normal in $G$. Then, for every $l\geq 1$, one has $\langle
c^{l}\rangle\cap\lambda\langle c^{l}\rangle\lambda^{-1}=\langle
c_{\lambda}^{l}\rangle$, which is infinite. Hence, every non-trivial subgroup
of $\langle c\rangle$ is s-normal in $\Lambda$.

Then, Lemma 8.15 implies that every non-trivial subgroup of $\langle
b^{k}\rangle=\langle c\rangle$ is s-normal in the amalgam $\Gamma$. Now,
$\langle c\rangle$ is cyclic, so it is not acylindrically hyperbolic by
Proposition 8.2, so $\Gamma$ is not acylindrically hyperbolic either by
Proposition 8.1.

(3) The group $\Gamma$ contains a copy of ${\rm BS}(m,n)$, which is non-linear
(see Remark 8.3). Hence $\Gamma$ cannot be a linear group. ∎

######  Remark 8.17.

The previous proposition shows in particular that if $|m|\neq 1$, $|n|\neq 1$,
$|m|\neq|n|$, and if one chooses $G$ such that $\langle c\rangle$ is not
highly core-free in $G$ (e.g. $G=\mathbb{Z}$ and $c\geq 2$), then results from
[MO15, HO16, FMS15, GGS20] do not apply to prove that $\Gamma$ is highly
transitive.

######  Remark 8.18.

On the other hand, if $|m|\neq 1$, $|n|\neq 1$, $|m|\neq|n|$, and if one
chooses $G$ such that $\langle c\rangle$ is highly core-free in $G$ (e.g.
$G={\rm BS}(m,n)$ and $c=b^{k}$), then Corollary B of [FMMM18] shows that
$\Gamma$ admits homogeneous actions on bounded Urysohn spaces.

###  8.2. Examples around finitely supported permutations

We now turn to examples constructed from the group of finite supported
permutations on an infinite countable set.

####  8.2.1. Examples of HNN extensions over $S_{f}(X)$

We denote by $S_{f}(X)$ the subgroup of $S(X)$ consisting in finitely
supported permutations.

The group $S_{f}(X)$ is known to be not linear but we could not find any
elementary proof in the literature and this is why we have chosen to include a
complete proof below. We thank Julien Bichon for explaining to us the
following argument.

######  Lemma 8.19.

Let $\Gamma$ be a group. If, for any prime number $q$ and any
$N\in\mathbb{N}^{*}$, $\Gamma$ contains a subgroup $G$ with
$G\simeq(\mathbb{Z}/q\mathbb{Z})^{N}$ then $\Gamma$ is not linear.

###### Proof.

Let $k$ be any algebraically closed field and denote by $p$ its
characteristic. Let us recall some elementary facts. For $n\in\mathbb{N}^{*}$,
let us denote by $U_{n}(k)\subset k^{*}$ the multiplicative subgroup of $n$-th
roots of unity. Elements of $U_{n}(k)$ are exactly the roots of the polynomial
$P=X^{n}-1\in k[X]$. Since $P^{\prime}=nX^{n-1}$ all the roots of $P$ are
simple if $p=0$ or if $p$ is a prime number which does not divide $n$. Hence,
if $p=0$ or $p$ is prime and does not divide $n$ one has $|U_{n}(k)|=n$. To
deduce the Lemma, it suffices to prove the following claim.

Claim. Let $V$ be a finite dimensional vector space over an algebraically
closed field $k$. If ${\rm GL}(V)$ contains a subgroup $G$ isomorphic to
$(\mathbb{Z}/q\mathbb{Z})^{N}$, where $N\in\mathbb{N}^{*}$ and $q$ is any
prime number with $q\neq{\rm char}(k)$ then $N\leq{\rm dim}(V)$.

Note that any element $g\in G$ satisfies $g^{q}=1$ hence, the minimal
polynomial $\mu_{g}$ of $g$ divides $X^{q}-1$. Since $q\neq{\rm char}(k)$,
$X^{q}-1$ has only simple roots so $\mu_{g}$ has only simple roots and $g$ is
diagonalizable with eigenvalues in $U_{q}(k)$. Moreover, since $G$ is finite
abelian and all its elements are diagonalisable, there exists a basis
$\mathcal{B}=(e_{1},\dots,e_{n})$ of $V$ which simultaneously diagonalises
every element of $G$. Let us denote by $\lambda(g)\in U_{q}(k)^{n}$ the
element $\lambda(g)=(\lambda_{1}(g),\dots,\lambda_{n}(g))$, where
$g(e_{k})=\lambda_{k}(g)e_{k}$. This defines an injective map $G\rightarrow
U_{q}(k)^{n}$, $g\mapsto\lambda(g)$. It follows that
$|G|=q^{N}\leq|U_{q}(k)^{n}|=q^{n}$, hence $N\leq n=\dim(V)$. ∎

######  Proposition 8.20.

The group $S_{f}(X)$ is not linear.

###### Proof.

Let $N\in\mathbb{N}^{*}$ and $q$ be a prime number. Since $X$ is infinite, one
can choose $q$-cycles $\sigma_{1},\ldots,\sigma_{N}$ in $S_{f}(X)$ with
pairwise disjoint supports. It is then easy to check that there is an
injective morphism of groups defined by

$(\mathbb{Z}/q\mathbb{Z})^{N}\to
S_{f}(X)\quad;\quad(x_{1},\ldots,x_{n})\mapsto\sigma_{1}^{x_{1}}\cdots\sigma_{N}^{x_{N}}\,.$

Hence, the proof follows from Lemma 8.19.∎

For any subset $F\subseteq X$, let us denote by $\Sigma(F)$ the pointwise
stabilizer of $F$ in $S_{f}(X)$, and remark that whenever $F$ is finite, the
subgroup $\Sigma(F)$ is infinite, in fact isomorphic to $S_{f}(X)$ itself. We
will abbreviate $\Sigma(\\{x\\})$ as $\Sigma(x)$. For any $k\geq 1$, let
$X^{(k)}$ denote the set of $k$-tuples of pairwise distinct points in $X$.

######  Lemma 8.21.

Let $F$ be a non-empty subset of $X$. The following hold:

  1. (1)

the stabilizer $\Sigma(F)$ is a core-free subgroup of $S_{f}(X)$, with
infinite index;

  2. (2)

if $F$ is finite, then $\Sigma(F)$ is not highly core-free in $S_{f}(X)$;

  3. (3)

if $F$ is finite, then $\Sigma(F)$ is s-normal in $S_{f}(X)$.

###### Proof.

(1) Let $x\in F$. Since the action $X\curvearrowleft S_{f}(X)$ is transitive
(even highly transitive) and faithful we have $\bigcap_{g\in
S_{f}(X)}g^{-1}\Sigma(x)g=\bigcap_{g\in S_{f}(X)}\Sigma(x\cdot
g)=\bigcap_{y\in X}\Sigma(y)=\\{1\\}$, hence $\Sigma(x)$ is core-free in
$S_{f}(X)$. A fortiori, $\Sigma(F)$ is core-free in $S_{f}(X)$. Let us denote
by $\tau_{y}\in S_{f}(X)$ is the transposition $\tau_{y}=(x\,\,\,y)$ for
$y\neq x$. The subgroup $\Sigma(x)$ has infinite index since, for all $y,z\in
X-\\{x\\}$, one has $\tau_{y}^{-1}\tau_{z}\in\Sigma(x)\Leftrightarrow y=z$. A
fortiori, $\Sigma(F)$ has infinite index in $S_{f}(X)$.

(2) Let us write $F=\\{x_{1},\ldots,x_{k}\\}$, with
$\bar{x}=(x_{1},\ldots,x_{k})\in X^{(k)}$. The action $X^{(k)}\curvearrowleft
S_{f}(X)$ is transitive, since the action $X\curvearrowleft S_{f}(X)$ is
highly transitive, and the stabilizer of $\bar{x}$ is $\Sigma(F)$.
Consequently, the action $\Sigma(F)\backslash S_{f}(X)\curvearrowleft
S_{f}(X)$ is conjugate to $X^{(k)}\curvearrowleft S_{f}(X)$. Now,
$X^{(k)}\curvearrowleft S_{f}(X)$ is not strongly faithful, since taking $k+1$
permutations with pairwise disjoint and finite supports in $X$, every point in
$X^{(k)}$ will be fixed by at least one of them. Consequently,
$\Sigma(F)\backslash S_{f}(X)\curvearrowleft S_{f}(X)$ is not highly faithful.

(3) For any $g\in S_{f}(X)$, we have $\Sigma(F)\cap
g^{-1}\Sigma(F)g=\Sigma(F\cup F\cdot g)$, and $F\cup F\cdot g$ is still
finite, hence $\Sigma(F)\cap g^{-1}\Sigma(F)g$ is infinite. ∎

######  Proposition 8.22.

Let $Y$ and $Z$ be two distinct infinite proper subsets of $X$, let $\tau:Y\to
Z$ be a bijection, and let $\vartheta=\tau_{*}:S_{f}(Y)\to S_{f}(Z)$ be the
isomorphism defined by $\vartheta(\sigma)=\tau^{-1}\sigma\tau$. Then, the HNN
extension $\Gamma=\operatorname{HNN}(S_{f}(X),S_{f}(Y),\vartheta)$ has the
following properties:

  1. (1)

it is not linear;

  2. (2)

it admits an action which is both highly transitive and highly faithful;

  3. (3)

if $Y$ and $Z$ are both cofinite, then it is not acylindrically hyperbolic;

  4. (4)

if $Y$ and $Z$ are both cofinite, then for every bounded subtree $\mathcal{B}$
of its Bass-Serre tree, the pointwise stabilizer $\Gamma_{\mathcal{B}}$ is not
highly core-free in a vertex stabilizer $\Gamma_{u}$.

###### Proof.

(1) Follows from Proposition 8.20.

(2) Note that $S_{f}(Y)=\Sigma(X-Y)$. Thus, $S_{f}(Y)$ is a core-free subgroup
of $S_{f}(X)$ by Lemma 8.21. Hence, Corollary F, or in this case Corollary
7.6, applies.

(3) As $Y$ and $Z$ are both cofinite and since the intersection of finitely
many cofinite subsets is cofinite, the powers $\tau^{n}$, defined by
composition of partial bijections in $X$ (for $n\in\mathbb{Z}$) all have a
cofinite domain and a cofinite range, that we will denote by $Y_{n}$ and
$Z_{n}$ respectively. Let $U$ be any cofinite subset of $X$. For any $g\in
S_{f}(X)$, one has $S_{f}(U)\cap g^{-1}S_{f}(U)g=S_{f}(U\cap U\cdot g)$, and
$U\cap U\cdot g$ is still cofinite. Moreover, for any $n\in\mathbb{Z}$, one
has $S_{f}(U)\cap t^{-n}S_{f}(U)t^{n}=S_{f}(U\cap U\tau^{n})$, where
$t\in\Gamma$ is the stable letter. Note that $U\tau^{n}$ is cofinite since the
bijection $\tau^{n}$ realizes a bijection between $U\cap Y_{n}$ and
$U\tau^{n}$, so that the subset $U\tau^{n}$ is cofinite in $Z_{n}$, hence
cofinite. Therefore, $U\cap U\tau^{n}$ is cofinite.

Then, given $\gamma\in\Gamma$, one can write
$\gamma=\gamma_{1}\cdots\gamma_{k}$, where each $\gamma_{j}$ is either a power
of the stable letter $t$, or an element of $S_{f}(X)$, and an easy induction
based on previous facts shows that $S_{f}(Y)\cap\gamma^{-1}S_{f}(Y)\gamma$
contains $S_{f}(V)$ for some cofinite set $V$. This proves that $S_{f}(Y)$ is
a s-normal subgroup in $\Gamma$.

Furthermore, $S_{f}(Y)$ is an amenable group, hence it is not acylindrically
hyperbolic by Proposition 8.2. Finally, Proposition 8.1 implies that $\Gamma$
is not acylindrically hyperbolic.

(4) Up to conjugating and to enlarging $\mathcal{B}$, we may and will assume
without loss of generality that the stabilizer $\Gamma_{u}$ is $S_{f}(X)$, and
that $\Gamma_{e}=S_{f}(Y)$ for some edge $e$ in $\mathcal{B}$. Since $\Gamma$
acts transitively on the positive edges, there exists
$\gamma_{1},\ldots,\gamma_{k}\in\Gamma$ such that

$\Gamma_{\mathcal{B}}=S_{f}(Y)\cap\bigcap_{j=1}^{k}\gamma_{j}^{-1}S_{f}(Y)\gamma_{j}\,.$

As in the proof of (2), we see that there exist cofinite sets
$V_{1},\ldots,V_{k}$ such that the intersection
$S_{f}(Y)\cap\gamma_{j}^{-1}S_{f}(Y)\gamma_{j}$ contains $S_{f}(V_{j})$ for
every $j$, hence $\Gamma_{\mathcal{B}}$ contains
$S_{f}(\bigcap_{j=1}^{k}V_{j})$, where $\bigcap_{j=1}^{k}V_{j}$ is cofinite.
Now, $S_{f}(\bigcap_{j=1}^{k}V_{j})$ is not highly core-free in
$\Gamma_{u}=S_{f}(X)$ by Lemma 8.21, hence $\Gamma_{\mathcal{B}}$ is not
either. ∎

######  Remark 8.23.

When both $Y$ and $Z$ are cofinite in $X$, this proposition provides more
explicit new examples of highly transitive groups, since items (1), (3) and
(4) show that the results from [MO15, HO16, FMS15, GGS20] do not apply.

######  Remark 8.24.

In the context of Proposition 8.22, notice that $\Gamma$ obviously admits a
highly transitive action when $\tau$ can be extended to a permutation
$\tilde{\tau}\in S(X)$. Indeed, the $\Gamma$-action defined by
$t\mapsto\tilde{\tau}$ and $\sigma\mapsto\sigma$ for $\sigma\in S_{f}(X)$ is
highly transitive since its restriction to $S_{f}(X)$ already is (in the
terminology of Section 3, this action corresponds to the global pre-action
$(X,\tilde{\tau})$). Nevertheless, the $\Gamma$-action we obtain factors
through the semi-direct product $S_{f}(X)\rtimes\langle\tilde{\tau}\rangle$,
which is amenable while $\Gamma$ is not; hence the $\Gamma$-action is not
faithful. Furthermore, such an extension to a permutation $\tilde{\tau}$ is
not possible when $X-Y$ and $X-Z$ have different cardinalities.

####  8.2.2. Examples of HNN extensions over
$S_{f}(\mathbb{Z})\rtimes\mathbb{Z}$

Let us now move to a modification of former examples to get groups which are
moreover finitely generated. For these examples, we consider the permutation
$s\in S_{f}(\mathbb{Z})$ given by $k\cdot s=k+1$. It is straightforward to
check that the subgroup $\langle S_{f}(\mathbb{Z}),s\rangle<S(\mathbb{Z})$ is
finitely generated, and isomorphic to a semi-direct product of the form
$S_{f}(\mathbb{Z})\rtimes\mathbb{Z}$. As before, for any subset
$F\subseteq\mathbb{Z}$, let us denote by $\Sigma(F)$ the pointwise stabilizer
of $F$ in $S_{f}(\mathbb{Z})$. For some purposes, we will need the action

$\mathbb{Z}\times\mathbb{Z}\curvearrowleft\langle
S_{f}(\mathbb{Z}),s\rangle\,,\qquad(k,l)\cdot
s^{n}g:=(ks^{n}g,ls^{n})=((k+n)g,l+n)$

for $g\in S_{f}(\mathbb{Z})$ and $n\in\mathbb{Z}$. Notice that this action is
faithful, as is the action $\mathbb{Z}\curvearrowleft\langle
S_{f}(\mathbb{Z}),s\rangle$. Moreover, given a subset $F\subseteq\mathbb{Z}$,
we observe that the pointwise stabilizer in $\langle
S_{f}(\mathbb{Z}),s\rangle$ of the subset
$F\times\\{0\\}\subset\mathbb{Z}\times\mathbb{Z}$ is the subgroup
$\Sigma(F)<S_{f}(\mathbb{Z})$.

######  Lemma 8.25.

Let $F$ be a non-empty subset of $\mathbb{Z}$. The following hold:

  1. (1)

the stabilizer $\Sigma(F)$ is a core-free subgroup of $\langle
S_{f}(\mathbb{Z}),s\rangle$, with infinite index;

  2. (2)

if $F$ is finite, then $\Sigma(F)$ is not highly core-free in $\langle
S_{f}(\mathbb{Z}),s\rangle$;

###### Proof.

(1) The group $\Sigma(F)$ is already core-free and has infinite index in
$S_{f}(\mathbb{Z})$ by Lemma 8.21.

(2) Let us write $F=\\{x_{1},\ldots,x_{k}\\}$, with $x_{1},\ldots,x_{k}$
pairwise distinct, and set

$\bar{x}=\big{(}(x_{1},0),\ldots,(x_{k},0)\big{)}\in(\mathbb{Z}\times\mathbb{Z})^{(k)}\,.$

Let us denote by $\Omega$ the orbit of $\bar{x}$ under $\langle
S_{f}(\mathbb{Z}),s\rangle$. As the action $\mathbb{Z}\curvearrowleft
S_{f}(\mathbb{Z})$ is highly transitive, $\Omega$ is the union
$\bigcup_{n\in\mathbb{Z}}(\mathbb{Z}\times\\{n\\})^{(k)}$. Furthermore, the
stabilizer of $\bar{x}$ is the pointwise stabilizer of $F\times\\{0\\}$, that
is, $\Sigma(F)$. Consequently, the action $\Sigma(F)\backslash\langle
S_{f}(\mathbb{Z}),s\rangle\curvearrowleft\langle S_{f}(\mathbb{Z}),s\rangle$
is conjugate to $\Omega\curvearrowleft\langle S_{f}(\mathbb{Z}),s\rangle$.
Now, $\Omega\curvearrowleft\langle S_{f}(\mathbb{Z}),s\rangle$ is not strongly
faithful, since taking $k+1$ elements of $S_{f}(\mathbb{Z})$ with pairwise
disjoint supports, every point in
$\Omega=\bigcup_{n\in\mathbb{Z}}(\mathbb{Z}\times\\{n\\})^{(k)}$ will be fixed
by at least one of them. Consequently, $\Sigma(F)\backslash\langle
S_{f}(\mathbb{Z}),s\rangle\curvearrowleft\langle S_{f}(\mathbb{Z}),s\rangle$
is not highly faithful.∎

Using Lemma 8.25 we can prove the following Proposition exactly as we proved
Proposition 8.22.

######  Proposition 8.26.

Let $Y$ and $Z$ be two distinct infinite proper subsets of $\mathbb{Z}$, let
$\tau:Y\to Z$ be a bijection, and let $\vartheta=\tau_{*}:S_{f}(Y)\to
S_{f}(Z)$ be the isomorphism defined by
$\vartheta(\sigma)=\tau^{-1}\sigma\tau$. Then, the HNN extension
$\Gamma=\operatorname{HNN}(\langle
S_{f}(\mathbb{Z}),s\rangle,S_{f}(Y),\vartheta)$ has the following properties:

  1. (1)

it admits an action which is both highly transitive and highly faithful;

  2. (2)

it is finitely generated and not linear;

  3. (3)

if $Y$ and $Z$ are both cofinite in $\mathbb{Z}$, then it is not
acylindrically hyperbolic;

  4. (4)

if $Y$ and $Z$ are both cofinite in $\mathbb{Z}$, then for every bounded
subtree $\mathcal{B}$ of its Bass-Serre tree, the pointwise stabilizer
$\Gamma_{\mathcal{B}}$ is not highly core-free in a vertex stabilizer
$\Gamma_{u}$.

######  Remark 8.27.

Again, when both $Y$ and $Z$ are cofinite in $\mathbb{Z}$, this proposition
provides explicit new examples of groups which are highly transitive.

Note that when the complements of $Y$ and $Z$ have the same cardinality, these
groups admit a natural highly transitive action, but it fails to be faithful.
Indeed, the subgroup $\langle S_{f}(\mathbb{Z}),t\rangle<\Gamma$ is isomorphic
to the HNN extension
$\operatorname{HNN}(S_{f}(\mathbb{Z}),S_{f}(Y),\vartheta)$, and the action of
this subgroup is not faithful by Remark 8.24.

####  8.2.3. Examples of amalgams

We now switch to the context of amalgams. We will need a refinement of Lemma
8.15.

######  Lemma 8.28.

Let us consider an amalgam $\Gamma=\Gamma_{1}*_{\Sigma}\Gamma_{2}$. Assume
there is a collection $\mathcal{C}$ of infinite subgroups of $\Sigma$ such
that, for every $\Sigma^{\prime}\in\mathcal{C}$ and every
$\gamma\in\Gamma_{1}\cup\Gamma_{2}$, the intersection
$\Sigma^{\prime}\cap\gamma^{-1}\Sigma^{\prime}\gamma$ contains an element of
$\mathcal{C}$. Then, for every $\Sigma^{\prime}\in\mathcal{C}$ and every
$\gamma\in\Gamma$, the intersection
$\Sigma^{\prime}\cap\gamma^{-1}\Sigma^{\prime}\gamma$ contains an element of
$\mathcal{C}$. In particular, all $\Sigma^{\prime}\in\mathcal{C}$ are s-normal
subgroups of $\Gamma$.

###### Proof.

Let $\Sigma_{0}$ be any element of $\mathcal{C}$, and let $\gamma$ be any
element of $\Gamma$, that we write as a product
$\gamma=\gamma_{1}\cdots\gamma_{n}$ of elements of $\Gamma_{1}$ or
$\Gamma_{2}$. Let us prove by induction that, for $k=0,\ldots,n$, there exists
$\Sigma_{k}\in\mathcal{C}$ which is contained in
$\Sigma_{0}\cap(\gamma_{1}\cdots\gamma_{k})^{-1}\Sigma_{0}(\gamma_{1}\cdots\gamma_{k})$.

For $k=0$, the group $\Sigma_{0}$ has been chosen in $\mathcal{C}$, and it
coincides with the intersection
$\Sigma_{0}\cap(\gamma_{1}\cdots\gamma_{k})^{-1}\Sigma_{0}(\gamma_{1}\cdots\gamma_{k})$
in this case. Then, for $k\geq 1$, the subgroup
$\Sigma_{k-1}\cap\gamma_{k}^{-1}\Sigma_{k-1}\gamma_{k}$ contains some
$\Sigma_{k}\in\mathcal{C}$. Moreover, one has
$\Sigma_{k-1}\subseteq\Sigma_{0}\cap(\gamma_{1}\cdots\gamma_{k-1})^{-1}\Sigma_{0}(\gamma_{1}\cdots\gamma_{k-1})$
by induction hypothesis, hence

$\displaystyle\Sigma_{k}$
$\displaystyle\subseteq\Sigma_{k-1}\cap\gamma_{k}^{-1}\Sigma_{k-1}\gamma_{k}$
$\displaystyle\subseteq\Sigma_{0}\cap(\gamma_{1}\cdots\gamma_{k-1})^{-1}\Sigma_{0}(\gamma_{1}\cdots\gamma_{k-1})\cap\gamma_{k}^{-1}\Sigma_{0}\gamma_{k}\cap(\gamma_{1}\cdots\gamma_{k})^{-1}\Sigma_{0}(\gamma_{1}\cdots\gamma_{k})$

whence
$\Sigma_{k}\subseteq\Sigma_{0}\cap(\gamma_{1}\cdots\gamma_{k})^{-1}\Sigma_{0}(\gamma_{1}\cdots\gamma_{k})$.

Finally, for $k=n$, we get
$\Sigma_{n}\subseteq\Sigma_{0}\cap\gamma^{-1}\Sigma_{0}\gamma$ with
$\Sigma_{n}\in\mathcal{C}$, as desired. ∎

######  Proposition 8.29.

Let $X,Y,Z$ be infinite countable sets such that $Z$ is proper subset of
$X\cap Y$. Then, the amalgam $\Gamma=S_{f}(X)*_{S_{f}(Z)}S_{f}(Y)$ has the
following properties:

  1. (1)

it admits an action which is both highly transitive and highly faithful;

  2. (2)

it is not a linear group;

  3. (3)

if $Z$ is cofinite in both $X$ and $Y$, then it is not acylindrically
hyperbolic;

  4. (4)

if $Z$ is cofinite in $X$ (resp. $Y$), then $S_{f}(Z)$ is not highly core-free
in $S_{f}(X)$ (resp. $S_{f}(Y)$).

###### Proof.

Lemma 8.21 and Corollary 7.7 imply (1). Proposition 8.20 implies (2) while (4)
follows from Lemma 8.21. Let us prove (3). Let $\mathcal{C}$ be the collection
of subgroups of the form $S_{f}(U)$ where $U$ is cofinite in $Z$ (hence
cofinite in both $X$ and $Y$). For every
$\Sigma^{\prime}=S_{f}(U)\in\mathcal{C}$ and every $\gamma\in S_{f}(X)\cup
S_{f}(Y)$, one can check as in former proofs that the intersection
$\Sigma^{\prime}\cap\gamma^{-1}\Sigma^{\prime}\gamma$ contains an element of
$\mathcal{C}$. Hence, Lemma 8.28 implies that $S_{f}(Z)$ is s-normal in
$\Gamma$. Since $S_{f}(Z)$ is amenable it is not acylindrically hyperbolic
hence $\Gamma$ is not acylindrically hyperbolic. ∎

Again, one can easily modify the previous examples to get groups which are
moreover finitely generated. The proof of the following Proposition is exactly
the same as the proof of Proposition 8.29 (by using Lemma 8.25). We can now
provide one last new class of highly transitive examples.

######  Proposition 8.30.

Let $Z$ be an infinite proper subset of $\mathbb{Z}$ and consider the amalgam
$\Gamma:=\langle S_{f}(\mathbb{Z}),s\rangle*_{S_{f}(Z)}\langle
S_{f}(\mathbb{Z}),s\rangle$. The following holds.

  1. (1)

$\Gamma$ admits an action which is both highly transitive and highly faithful;

  2. (2)

$\Gamma$ is finitely generated and not linear;

  3. (3)

If $Z$ is cofinite in $\mathbb{Z}$, then $\Gamma$ is not acylindrically
hyperbolic and $S_{f}(Z)$ is not highly core-free in $\langle
S_{f}(\mathbb{Z}),s\rangle$.

###  8.3. Faithful actions which are non-topologically free on the boundary

Although our main result provides a complete characterization of high
transitivity for groups admitting a faithful minimal action of general type on
a tree, one may wonder if Corollaries 7.6 and 7.7 can hold in a wider context,
namely, if the core-freeness assumption of the edge group in a vertex group
can be weakened to core-freeness in the whole group. We will see that it is
not the case.

Thanks to the quoted result from [LBMB19], this amounts to finding examples of
amalgams and HNN extensions whose action on their Bass-Serre trees are minimal
of general type and faithful, but the action on the boundary is not
topologically free. By Bass-Serre theory, we essentially need to find faithful
edge transitive actions on trees without inversions which are not
topologically free on the boundary, but which in the amalgam case have two
vertex orbits, while in the HNN case they have only one vertex orbit.

Our examples belong to a class which was explored in depth by Le Boudec [LB16,
LB17], generalizing a construction of Bader-Caprace-Gelander-Mozes [BCGM12]
which takes its roots in the work of Burger-Mozes [BM00]. Such examples
already appeared in Le Boudec and Matte-Bon’s work on high transitivity, so
our only contribution here is to point out that some of those naturally
decompose as amalgams or HNN extensions. We will focus on specific easy
examples instead of seeking large generality. For more examples, we refer the
reader to Ivanov’s recent work [Iva20].

####  8.3.1. An example of amalgam

Let $\mathcal{T}_{d}$ be a $d$-regular tree of finite degree $d\geq 3$. As in
[BM00], let us fix a coloring on the set of edges
$c:E(\mathcal{T}_{d})\to\\{1,\ldots,d\\}$ such that:

  * •

every edge has the same color as its antipode;

  * •

for any vertex $v$, the restriction of $c$ to the star
$\operatorname{\mathrm{st}}(v)$ is a bijection onto $\\{1,\ldots,d\\}$.

For any vertex $v$, any automorphism $g\in\operatorname{Aut}(\mathcal{T}_{d})$
induces a bijection
$g_{v}:\operatorname{\mathrm{st}}(v)\to\operatorname{\mathrm{st}}(gv)$, which
itself induces a permutation $\sigma(g,v)\in S_{d}$, where
$S_{d}=\operatorname{Sym}(\\{1,\ldots,d\\})$. Let $C_{d}$ be a cyclic subgroup
of $S_{d}$ generated by a $d$-cycle. Then, the group, coming from [LB16],

$G=G(C_{d})=\\{g\in\operatorname{Aut}(\mathcal{T}_{d}):\,\sigma(g,v)\in
C_{d}\text{ for all but finitely many vertices}\\}$

is countable (indeed, note that any automorphism $g\in G(C_{d})$ is completely
determined by the restriction of $\sigma(g,\cdot)$ to the set of vertices $v$
such that $\sigma(g,v)\not\in C_{d}$).

In order to forbid inversions, recall there is a natural equivalence relation
$\mathcal{R}_{even}$ on $V(\mathcal{T}_{d})$ which relates any two vertices at
even distance from each other, and that this equivalence relation is preserved
by any automorphism of $\mathcal{T}_{d}$. We then let

$\Gamma=G^{+}=\\{g\in G:\,g\text{ does not exchange the two classes of
}\mathcal{R}_{even}\\}\,.$

It is fairly easy to see that the action
$\Gamma\curvearrowright\mathcal{T}_{d}$ is transitive on undirected edges,
hence minimal, of general type, and without inversion. Let us now fix some
edge $e_{0}$ from $v_{1}$ to $v_{2}$, and consider the stabilizers
$\Gamma_{1}$, $\Gamma_{2}$ and $\Sigma$ of $v_{1}$, $v_{2}$ and $e_{0}$
respectively (in $\Gamma$). By Bass-Serre theory, we have the following.

######  Remark 8.31.

The morphism $\Gamma_{1}\ast_{\Sigma}\Gamma_{2}\to\Gamma$ given by inclusions
is an isomorphism, and $\mathcal{T}_{d}$ is the Bass-Serre tree of
$\Gamma_{1}\ast_{\Sigma}\Gamma_{2}$.

The following result summarizes well-known properties of $\Gamma$ showing that
the hypothesis that $\Gamma\curvearrowright\partial\mathcal{T}$ is
topologically free cannot be relaxed in Corollary 7.7. We provide a proof for
the reader’s convenience.

######  Proposition 8.32.

With the above notations:

  1. (1)

$\Sigma$ is core-free in $\Gamma$, and the amalgam
$\Gamma_{1}\ast_{\Sigma}\Gamma_{2}$ is non-degenerate;

  2. (2)

the $\Gamma$-action on $\partial\mathcal{T}_{d}$ is not topologically free;

  3. (3)

$\Gamma$ is not highly transitive;

  4. (4)

$\Gamma$ is icc.

###### Proof.

(1) The $\Gamma$-action on $\mathcal{T}_{d}$ is faithful (by definition) and
of general type, so that $\Sigma$ is core-free in $\Gamma$, and the amalgam
$\Gamma_{1}\ast_{\Sigma}\Gamma_{2}$ is non-degenerate.

(2) Consider the half-tree $\mathcal{H}$ associated to $e_{0}$. It suffices to
prove that the pointwise stabilizer of $\mathcal{H}$ is a non-trivial group.

To do so, we follow the proof of [LB17, Theorem C]. First, we take a non-
trivial permutation $\sigma\in S_{d}$ which fixes the color of $e_{0}$ (this
exists since $d\geq 3$; notice it lives in $S_{d}-C_{d}$). Then, we define a
non-trivial automorphism $\gamma\in\Gamma$ fixing $\mathcal{H}$ pointwise as
follows.

  * •

The restriction of $\gamma$ to $\mathcal{H}$ is the identity.

  * •

Then, we let $\gamma$ act on $\operatorname{\mathrm{st}}(v_{1})$ so that
$\sigma(g,v_{1})=\sigma$ (this is possible since $\sigma$ fixes the color of
$e_{0}$, and will guarantee that $\gamma$ is non-trivial since $\sigma$ is
non-trivial).

  * •

Then, we extend the action inductively: given any vertex $w$ outside
$\mathcal{H}$, we set $w^{\prime}$ to be the unique neigbour of $w$ which is
closer to $\mathcal{H}$ than $w$, and define the $\gamma$-action on
$\operatorname{\mathrm{st}}(w)$ in terms of the (previously defined)
$\gamma$-action on $\operatorname{\mathrm{st}}(w^{\prime})$. Namely, denoting
by $e_{w}$ the edge from $w$ to $w^{\prime}$, the $\gamma$-action on
$\operatorname{\mathrm{st}}(w^{\prime})$ provides the edge $\gamma e_{w}$.
Then there is a unique element $\sigma_{w}\in C_{d}$ sending $c(e_{w})$ onto
$c(\gamma e_{w})$, and we let $\gamma$ act on $\operatorname{\mathrm{st}}(w)$
so that $\sigma(g,w)=\sigma_{w}$ (note that $\sigma(g,w)=\sigma(g,w^{\prime})$
as soon as $\sigma(g,w^{\prime})$ was already in $C_{d}$).

(3) The $\Gamma$-action on $\mathcal{T}_{d}$ is minimal, by edge-transitivity,
and of general type. Consequently, [LBMB19, Corollary 1.5] applies, and the
transitivity degree of $\Gamma$ is at most $2$.

(4) Let $\gamma_{0}$ be a non-trivial element of $\Gamma$ and $\xi$ be a point
in $\partial\mathcal{T}_{d}$ which is not fixed by $\gamma_{0}$. Given edges
$e,e^{\prime}$ with sources in the same class of vertices, such that $e$ is on
the geodesic $[\xi,s(e^{\prime})]$, there exists $\gamma\in\Gamma$ such that
$\gamma e=e^{\prime}$. This $\gamma$ is a hyperbolic element whose axis
contains $e$ and $e^{\prime}$. Moreover, if we choose $e$ close enough to
$\xi$, then the repelling point $\xi^{-}$ of $\gamma$ in
$\partial\mathcal{T}_{d}$ is close enough to $\xi$ so that
$\gamma_{0}\xi^{-}\neq\xi^{-}$. Now, since $\xi^{-}$ is not fixed by
$\gamma_{0}$, the set of fixed points of $\gamma^{n}\gamma_{0}\gamma^{-n}$
moves into smaller and smaller neighborhoods (in
$\mathcal{T}_{d}\cup\partial\mathcal{T}_{d}$) of the attracting point of
$\gamma$ as $n\to+\infty$. Thus, the set
$\\{\gamma^{n}\gamma_{0}\gamma^{-n}:\,n\geq 1\\}$ is infinite. ∎

####  8.3.2. An example of HNN extension

In the previous example, notice that even the subgroup

$U(\mathord{\operatorname{id}})=\\{g\in\operatorname{Aut}(\mathcal{T}_{d}):\,\sigma(g,v)=\mathord{\operatorname{id}}\text{
for every vertex }v\\}$

includes inversions, so that one cannot easily find a subgroup of $G$ without
inversion and acting transitively on $V(\mathcal{T}_{d})$. Hence, we slightly
modify the construction in order to get an example of HNN extension.

Let $\mathcal{T}_{d,d}$ be a $(d,d)$-biregular tree, where $d\geq 2$ (by this,
we mean an oriented tree in which every star $\operatorname{\mathrm{st}}(v)$
contains exactly $d$ positive edges and $d$ negative edges). Let us denote
$\operatorname{\mathrm{st}}(v)^{+}=\operatorname{\mathrm{st}}(v)\cap
E(\mathcal{T}_{d,d})^{+}$ and
$\operatorname{\mathrm{st}}(v)^{-}=\operatorname{\mathrm{st}}(v)\cap
E(\mathcal{T}_{d,d})^{-}$, and fix a coloring on the set of edges
$c:E(\mathcal{T}_{d,d})\to\\{1,\ldots,2d\\}$ such that:

  * •

every edge has the same color as its antipode;

  * •

for any vertex $v$, the restriction of $c$ to the star
$\operatorname{\mathrm{st}}(v)$ is a bijection onto $\\{1,\ldots,2d\\}$;

  * •

for any vertex $v$, the image $c(\operatorname{\mathrm{st}}(v)^{+})$ is either
$\\{1,\ldots,d\\}$ or $\\{d+1,\ldots,2d\\}$.

By $\operatorname{Aut}(\mathcal{T}_{d,d})$, we mean the group of automorphisms
of $\mathcal{T}_{d,d}$ preserving the orientation. For any vertex $v$, any
automorphism $g\in\operatorname{Aut}(\mathcal{T}_{d,d})$ induces bijections
$g_{v}^{\pm}:\operatorname{\mathrm{st}}(v)^{\pm}\to\operatorname{\mathrm{st}}(gv)^{\pm}$,
which themselves induces a permutation $\sigma(g,v)\in S_{2d}$, where
$S_{2d}=\operatorname{Sym}(\\{1,\ldots,2d\\})$, preserving the partition
$\\{1,\ldots,d\\}\sqcup\\{d+1,\ldots,2d\\}$. Let $F_{d}$ be a the subgroup of
$S_{2d}$ generated by the commuting elements

$\displaystyle\sigma_{1}$ $\displaystyle=(1\quad 2\quad\cdots\quad d)(d+1\quad
d+2\quad\cdots\quad 2d),$ $\displaystyle\sigma_{2}$ $\displaystyle=(1\quad
d+1)(2\quad d+2)\cdots(d\quad 2d).$

Then, the group,

$\Gamma=G(F_{d})=\\{g\in\operatorname{Aut}(\mathcal{T}_{d,d}):\,\sigma(g,v)\in
F_{d}\text{ for all but finitely many vertices}\\}$

is countable (this is not hard to prove, using that $F_{d}$ acts freely on
$\\{1,\ldots,2d\\}$).

It is fairly easy to see that the action
$\Gamma\curvearrowright\mathcal{T}_{d,d}$ is transitive on positive edges,
hence minimal, of general type. It is moreover without inversion since it
preserves the orientation. Let us fix some vertex $v$, some positive edges
$e_{1},e_{2}$ such that $r(e_{1})=v=s(e_{2})$, and some automorphism
$\tau\in\Gamma$ such that $\tau(e_{1})=e_{2}$. Now, consider the stabilizers
$H=\Gamma_{v}$ and $\Sigma=\Gamma_{e_{2}}$, and the isomorphism
$\vartheta:\Sigma\to\Gamma_{e_{1}}$ given by
$\vartheta(\sigma)(x)=\tau^{-1}\sigma\tau(x)$. By Bass-Serre theory, we have
the following.

######  Remark 8.33.

The morphism $\operatorname{HNN}(H,\Sigma,\vartheta)\to\Gamma$ given by
inclusions is an isomorphism, and $\mathcal{T}_{d,d}$ is the Bass-Serre tree
of $\operatorname{HNN}(H,\Sigma,\vartheta)$.

The following result summarizes well-known properties of $\Gamma$ showing that
the hypothesis that $\Gamma\curvearrowright\partial\mathcal{T}$ is
topologically free cannot be relaxed in Corollary 7.6. We omit the proof,
which is similar to the one of Proposition 8.32.

######  Proposition 8.34.

With the above notations:

  1. (1)

$\Sigma$ is core-free in $\Gamma$, and the HNN extension
$\operatorname{HNN}(H,\Sigma,\vartheta)$ is non-ascending;

  2. (2)

the $\Gamma$-action on $\partial\mathcal{T}_{d,d}$ is not topologically free;

  3. (3)

$\Gamma$ is not highly transitive;

  4. (4)

$\Gamma$ is icc.

##  9\. Other types of actions and necessity of the minimality assumption

We now discuss various natural extensions of Theorem A by considering other
types of actions (recall that group actions on trees are classified in five
different types, see Section 2.3). As we will see, non-general type actions
which are topologically free on the boundary seem to play no role regarding
high transitivity. We will also see that the minimality hypothesis in Theorem
A cannot be avoided.

######  Proposition 9.1.

Every infinite residually finite group admits an elliptic faithful action on a
tree such that the action on the boundary is both free and minimal.

###### Proof.

Let $\Gamma$ be a residually finite group, let $\Gamma_{n}$ be a decreasing
chain of finite index normal subgroups with trivial intersection, where
$\Gamma_{0}=\Gamma$. Then the disjoint union of the coset spaces
$\Gamma/\Gamma_{n}$ has a natural tree structure where we connect each
$\gamma\Gamma_{n+1}$ to $\gamma\Gamma_{n}$. Since the action is transitive on
each level of the tree, this action is minimal on the boundary. It is free
because the subgroups $\Gamma_{n}$ are normal and intersect trivially. Finally
it is elliptic because the vertex $\Gamma$ is fixed (it is moreover the only
non-trivial invariant subtree). ∎

Since there are both highly transitive residually finite groups (such as
$\mathbb{F}_{2}$) and non-highly transitive residually finite groups (such as
$\mathbb{Z}$), we see that there is no hope for a classification of the
transitivity degree of groups admitting an elliptic faithful action on a tree
which is free and minimal on the boundary. We can use also this construction
in order to show that the minimality assumption is needed in Theorem A.

######  Proposition 9.2.

Let $\Gamma$ be a group endowed with a faithful action of general type on a
tree. Then for every residually finite group $\Lambda$, the group
$\Gamma\times\Lambda$ admits an action of general type on a tree which is
topologically free on the boundary.

###### Proof.

Let $\mathcal{T}_{1}$ be a tree admitting a faithful $\Gamma$-action of
general type. Let $\Lambda\curvearrowright\mathcal{T}_{2}$ be an action
provided by the previous proposition, let $o$ be its unique fixed point. Our
new tree $\mathcal{T}$ is obtained by gluing over each vertex of
$\mathcal{T}_{1}$ a copy of $\mathcal{T}_{2}$ at its origin $o$. To be more
precise, the vertex set is $V(\mathcal{T})=V(\mathcal{T}_{1})\times
V(\mathcal{T}_{2})$, and on the vertex set $\mathcal{T}_{1}\times\\{o\\}$ we
put a copy of the edges of $\mathcal{T}_{1}$, while for each $v\in
V(\mathcal{T}_{1})$, we put a copy of the edges of $\mathcal{T}_{2}$ on the
vertex set $\\{v\\}\times\mathcal{T}_{2}$.

Then the $\Gamma\times\Lambda$ action on $V(\mathcal{T})$ given by
$(\gamma,\lambda)\cdot x=(\gamma\cdot x,\lambda\cdot x)$ is an action by
automorphisms on our new tree $\mathcal{T}$. Noting that each half-tree
contains a copy of $\mathcal{T}_{2}$, it is not hard to check that this action
is moreover topologically free. Moreover, it is of general type since the
$\Gamma$-action on $\mathcal{T}_{2}$ was of general type. ∎

This proposition yields many examples of non-highly transitive groups thanks
to the following results.

######  Proposition 9.3.

Let $G$ be a non-abelian topologically simple group. If $\Gamma$ is a dense
subgroup of $G$, then the centralizer of every non-trivial element of $\Gamma$
is core-free.

###### Proof.

Let $\gamma\in\Gamma\setminus\\{1_{\Gamma}\\}$. If the centralizer of $\Gamma$
is not core-free, then there exists a normal subgroup $N\leq\Gamma$ such that
every element of $N$ commutes with $\gamma$. Since $G$ is topologically
simple, $N$ is dense in $G$, so by continuity of group multiplication we
conclude that every element of $G$ commutes with $\gamma$. In particular, $G$
has a non-trivial center, which contradicts the topological simplicity of $G$
since $G$ is not abelian. ∎

######  Corollary 9.4.

Let $\Gamma$ be a highly transitive group. Then the centralizer of every non-
trivial element of $\Gamma$ must be core-free, and $\Gamma$ cannot be
decomposed as a non-trivial direct product.

###### Proof.

If $\Gamma$ is highly transitive, it can be embedded as a dense subgroup of
$S(X)$ for some infinite set $X$, and since the latter is topologically
simple, the conclusion follows from the previous result. Moreover, if $\Gamma$
could be decomposed as a non-trivial direct product
$\Gamma_{1}\times\Gamma_{2}$, then if
$\gamma\in\Gamma_{1}\setminus\\{1_{\Gamma_{1}}\\}$, the element
$(\gamma,1_{\Gamma_{2}})$ would commute with every element of the non-trivial
normal subgroup $\\{1_{\Gamma_{1}}\\}\times\Gamma_{2}$, a contradiction. ∎

Applying the above proposition for instance to $\Gamma={\rm BS}(2,3)$ and
$\Lambda=\mathbb{Z}$, we see that the group $BS(2,3)\times\mathbb{Z}$ admits a
(faithful) action of general type on a tree $\mathcal{T}$ which is
topologically free on the boundary, although it is not highly transitive
because it decomposes as a direct product. We see moreover that when
restricting to the minimal component of this action of a finitely generated
group, we will loose the faithfulness of the action (in particular the
topological freeness), which is why Theorem A cannot be applied.

We now move on to showing that no general classification can be hoped for in
the case of parabolic actions (note that minimal parabolic actions only arise
for non-finitely generated groups).

######  Lemma 9.5.

Every non-finitely generated group admits a faithful parabolic action on a
tree.

###### Proof.

Since $\Gamma$ is countable, it can be written as a countable increasing union
of finitely generated subgroups $\Gamma=\bigcup_{n\in\mathbb{N}}\Gamma_{n}$,
where $\Gamma_{0}=\\{1\\}$. We now put a tree structure on the vertex set
$\bigsqcup_{n}\Gamma/\Gamma_{n}$ by connecting each $\gamma\Gamma_{n}$ to
$\gamma\Gamma_{n+1}$. $\Gamma$ acts on this tree by left translation. Since
$\Gamma_{0}=\\{1\\}$, this action is faithful. Note that every group element
$g\in\Gamma$ is elliptic (with fixed point $\Gamma_{n}$, where
$n\in\mathbb{N}$ is such that $g\in\Gamma_{n}$). Since $\Gamma$ is not
finitely generated, this action has no global fixed point, so we have a
parabolic action. ∎

Note that the $\Gamma$-invariant subtrees of the action constructed above are
exactly those of the form $\sqcup_{n\geq m}\Gamma/\Gamma_{n}$ for some $m\geq
0$. So this action has no minimal globally invariant subtree, as opposed to
what happens for finitely generated groups.

######  Proposition 9.6.

Let $\Gamma$ be a non-finitely generated group, let $\Lambda$ be a residually
finite infinite group. Then $\Gamma\times\Lambda$ admits a parabolic action on
a tree such that the action on the boundary is topologically free.

###### Proof.

As before we use the $\Gamma$-action on a tree $\mathcal{T}_{1}$ provided by
the previous lemma, and $\Lambda\curvearrowright\mathcal{T}_{2}$ be an
elliptic action provided by Proposition 9.1, whose unique fixed point is
denoted by $o\in V(\mathcal{T}_{2})$. This time, we glue a copy of
$\mathcal{T}_{2}$ on each vertex at the level $0$ of the tree
$\mathcal{T}_{1}$, thus yielding a $\Gamma\times\Lambda$ action which is
easily seen to be elliptic. Moreover, the action on the boundary is
topologically free. Note furthermore that the action on the boundary has a
unique fixed point, corresponding to the unique element of the boundary of
$\mathcal{T}_{1}$. ∎

As an example, we can take for $\Gamma$ the group of finitely supported
permutations, and for $\Lambda$ the group $\mathbb{Z}$, and we get a non-
highly transitive group with a parabolic action which is topologically free on
the boundary, and which has no minimal component. We do not know if there is a
highly transitive group with a parabolic action which is topologically free on
the boundary.

Let us now treat the quasi-parabolic case. First, note that the Baumslag-
Solitar groups ${\rm BS}(1,n)$ for $n\geq 2$ provide examples of groups
admitting an action on a tree which is minimal and quasi-parabolic (since it
is an ascending HNN extension, cf. Sec. 2.5) and topologically free on the
boundary (by Lemma 8.6), but which are not highly transitive since they are
solvable.

######  Remark 9.7.

Another example of a non-highly transitive group with a quasi-parabolic
minimal action on a tree which is topologically free on the boundary is
provided by Thompson’s group $F=\left\langle
x_{0},x_{1},x_{2},\ldots|\,x_{k}^{-1}x_{n}x_{k}=x_{n+1}\text{ for all
}k<n\right\rangle$. $F$ is not highly transitive since, by [LBMB19, Corollary
5.3], it has transitivity degree at most $2$. Let $H$ be the subgroup
generated by $\\{x_{i}:i\geq 1\\}$ (which is isomorphic to $F$), and
$\vartheta$ be the endomorphism which takes $x_{i}$ to $x_{i+1}$ and observe
that $F=\operatorname{HNN}(H,H,\vartheta)$. By Section 2.5 the action of $F$
on the associated Bass-Serre tree is quasi-parabolic and minimal. Moreover,
since $x_{0}^{-k}\vartheta(H)x_{0}^{k}=\langle x_{n}\,|\,n\geq k+2\rangle$ for
all $k\geq 1$, it is not difficult to check that $\vartheta(H)$ is core-free
in $H$ hence, the action is also topologically free on the boundary.

Coupled with the previous examples, the following proposition shows that for
groups admitting minimal quasi-parabolic actions on a tree, the topological
freeness of the action does not play a role in their high transitivity.

######  Proposition 9.8.

The finitely generated group $\Gamma=S_{f}(\mathbb{Z})\rtimes\mathbb{Z}$ is
highly transitive and finitely generated but admits a minimal quasi-parabolic
action on a tree which is topologically free on the boundary.

###### Proof.

We have already observed that $\Gamma$ is highly transitive thanks to its
natural action on $\mathbb{Z}$. We will first show that it can be written as
an ascending HNN extension. Denote by $\tau$ the translation on $\mathbb{Z}$.
Let $\vartheta$ be the corresponding inner automorphism of $S(\mathbb{Z})$,
i.e. $\vartheta(\gamma)=\tau^{-1}\gamma\tau$.

Consider the subgroups $H=\Sigma=S_{f}(\mathbb{N})$ which we view as subgroups
of $S_{f}(\mathbb{Z})<\Gamma$. Note that $\vartheta(\Sigma)$ is the stabilizer
of $0$ in $S_{f}(\mathbb{N})$ i.e.
$\vartheta(\Sigma)=\Sigma(0)<S_{f}(\mathbb{N})$ with the notations of Section
8.2. We claim that $\Gamma=\operatorname{HNN}(H,\Sigma,\vartheta)$.

First, $\Gamma$ clearly satisfies the defining relations of the HNN extension
$\operatorname{HNN}(H,\Sigma,\vartheta)$, and since it is generated by $H$ and
$t$, it is a quotient of $\operatorname{HNN}(H,\Sigma,\vartheta)$ via a
canonical map $\pi:\operatorname{HNN}(H,\Sigma,\vartheta)\to\Gamma$. We need
to show that $\pi$ is injective.

In order to do so, we fix a set of representatives $C^{-}$ of left
$\vartheta(\Sigma)$-cosets in $H$, such as $C^{-}=\\{(0\quad i):i\geq 1\\}$.
Let $\gamma\in\operatorname{HNN}(H,\Sigma,\vartheta)\setminus\\{1\\}$, we need
to show $\pi(\gamma)\neq 1_{\Gamma}$. Since $\Sigma=H$, the normal form
described in Section 2.5 for $\gamma$ is

$\gamma=c_{1}t^{\varepsilon_{1}}\cdots c_{n}t^{\varepsilon_{n}}h_{n+1},$

where $n\geq 0$, $\varepsilon_{i}=\pm 1$ for $1\leq i\leq n$,
$\varepsilon_{i}=+1$ implies $c_{i}=1$, $\varepsilon_{i}=-1$ implies $c_{i}\in
C^{-}$, $h_{n+1}\in H$, and there is no subword of the form
$t^{\varepsilon}1t^{-\varepsilon}$. Note that this implies that if
$\varepsilon_{i}=-1$, then $\varepsilon_{j}=-1$ for all $j\geq i$.

The case $n=0$ is the case $\gamma\in H$, for which $\pi(\gamma)=\gamma$, so
we may as well assume $n\geq 1$. Denote by $\varepsilon:\Gamma\to\mathbb{Z}$
the morphism coming from the decomposition of $\Gamma$ as a semi-direct
product $S_{f}(\mathbb{Z})\rtimes\mathbb{Z}$. Note that

$\varepsilon[\pi(\gamma)]=\sum_{i=1}^{n}\varepsilon_{i}.$

We now have two cases to consider for the normal form of
$\gamma\in\operatorname{HNN}(H,\Sigma,\vartheta)\setminus\\{1\\}$.

  * •

If $\varepsilon_{1}=-1$, then all the $\varepsilon_{i}$ must be equal to $-1$,
so $\varepsilon(\pi(\gamma))\neq 0$, in particular $\pi(\gamma)\neq
1_{\Gamma}$.

  * •

If $\varepsilon_{1}=+1$, then the normal form of $\gamma$ is

$\gamma=1t1t\cdots 1tc_{i+1}t^{-1}\cdots
c_{n}t^{-1}h_{n+1}=t^{i}c_{i+1}t^{-1}\cdots c_{n}t^{-1}h_{n+1}.$

Since $\varepsilon[\pi(\gamma)]=-i+(n-i)$, we only need to consider the case
$n=2i$. Observe that for all $c\in C^{-}$ we have $\tau
c\tau^{-1}=\vartheta^{-1}(c)$. We then have

$\displaystyle\pi(t^{i}c_{i+1}t^{-1}\cdots c_{n}t^{-1}h_{n+1})$
$\displaystyle=\tau^{i}c_{i+1}\tau^{-1}\cdots c_{n}\tau^{-1}h_{n+1}$
$\displaystyle=\vartheta^{-1}\left(\vartheta^{-1}(\cdots(\vartheta^{-1}(\vartheta^{-1}(c_{i+1})c_{i+2})c_{i+3})\cdots)c_{2i}\right)h_{n+1}$

A straightforward induction on the number of iterates of $\vartheta^{-1}$ in
the above expression then shows that
$\vartheta^{-1}\left(\vartheta^{-1}(\cdots(\vartheta^{-1}(\vartheta^{-1}(c_{i+1})c_{i+2})c_{i+3})\cdots)c_{2i}\right)$
does not fix the element $-i$. Since $h_{n+1}$ fixes all the negative
elements, we conclude that $\pi(\gamma)$ does not fix $-i$, and hence it is
not trivial.

So we do have $\Gamma=\operatorname{HNN}(H,\Sigma,\vartheta)$, in particular
it is an ascending non-degenerate HNN extension. So as explained in Section
2.5, its action on its Bass-Serre tree is minimal and quasi-parabolic.
Finally, $\vartheta(\Sigma)=\Sigma(0)$ is core-free in $H=S_{f}(\mathbb{N})$
by Lemma 8.21, so by Lemma 7.3 we conclude that the action on the boundary is
topologically free as wanted. ∎

We finally mention the lineal case. Note that in this case, minimal actions
are not interesting with respect to high transitivity since no subgroup of the
automorphism group of the biinfinite line is highly transitive.

In the elliptic case, a natural weakening of the minimality assumption was
provided by asking that the action on the boundary is minimal. Here however,
this is still too weak a condition since any lineal action will have the two
ends corresponding to the axis as an invariant set. We thus replace it by
topological transitivity and observe that in this setup, there seems to be no
connection between high transitivity and lineal actions.

######  Proposition 9.9.

Let $\Gamma$ be a residually finite group, then the group
$\Gamma\times\mathbb{Z}$ admits a lineal action on a tree which is both
topologically free and topologically transitive on the boundary.

###### Proof.

Let $\mathcal{T}$ be a tree equipped with an elliptic $\Gamma$-action which is
minimal on the boundary, let $o$ be the fixed point. As in the previous
constructions, we then glue a copy of $\mathcal{T}$ at the vertex $o$ on top
of every element of $\mathbb{Z}$, thus obtaining a tree with a natural
$\Gamma\times\mathbb{Z}$-action which is minimal on the boundary. ∎

The previous proposition provides us many non-highly transitive groups with a
lineal action on a tree which is both topologically free and topologically
transitive on the boundary.

In the opposite direction, the group $S_{f}(\mathbb{Z})\rtimes\mathbb{Z}$
provides us an example of a highly transitive group satisfying the assumptions
of the previous proposition, thus showing that lineal actions which are
topologically free and topologically transitive on the boundary do not play a
role in high transitivity.

######  Proposition 9.10.

The highly transitive group $S_{f}(\mathbb{Z})\rtimes\mathbb{Z}$ admits a
lineal action on a tree which is both topologically free and topologically
transitive on the boundary.

###### Proof.

Denote by $\vartheta$ the action of $\mathbb{Z}$ on $S_{f}(\mathbb{Z})$ by
shift which yields the direct product $S_{f}(\mathbb{Z})\rtimes\mathbb{Z}$.
Consider the tree $\mathcal{T}$ whose vertex set

$V(\mathcal{T})=\\{0\\}\times\\{*\\}\times\mathbb{Z}\sqcup\mathbb{N}^{*}\times
S_{f}(\mathbb{Z})\times\mathbb{Z}$

equipped with the following edges structure: for all $n\in\mathbb{N}$,
$m\in\mathbb{Z}$ and $\sigma\in S_{f}(\mathbb{Z})$

  * •

we have an edge from $(0,*,m)$ to $(0,*,m+1)$;

  * •

we have an edge from $(0,*,m)$ to $(1,\sigma,m)$;

  * •

we have an edge from $(n,\sigma,m)$ to $(n+1,\sigma,n)$.

Then consider the left action of $S_{f}(\mathbb{Z})\rtimes\mathbb{Z}$ on
$V(\mathcal{T})$ given by $(\sigma,k)\cdot(0,*,m)=(0,*,m+k)$ and
$(\sigma,k)\cdot(n,\tau,m)=(n,\sigma\vartheta^{k}(\tau),m+k)$. It is not hard
to check that this action is both topologically free and topologically
transitive on the boundary. ∎

## References

  * [BCGM12] Uri Bader, Pierre-Emmanuel Caprace, Tsachik Gelander, and Shahar Mozes.  Simple groups without lattices.  Bulletin of the London Mathematical Society, 44(1):55–67, February 2012.  doi:10.1112/blms/bdr061.
  * [BIO19] Rasmus Sylvester Bryder, Nikolay Ivanov, and Tron Omland.  C*-simplicity of HNN extensions and groups acting on trees.  to appear in Ann. Inst. Fourier, April 2019.  arXiv:1711.10442.
  * [BM00] Marc Burger and Shahar Mozes.  Groups acting on trees: From local to global structure.  Publications mathématiques de l’IHÉS, 92(1):113–150, December 2000.  URL: http://link.springer.com/10.1007/BF02698915, doi:10.1007/BF02698915.
  * [Cha12] Vladimir Chaynikov.  Properties of hyperbolic groups: free normal subgroups, quasi- convex subgroups and actions of maximal growth. PhD thesis, Vanderbilt University, 2012.
  * [DGO17] F. Dahmani, V. Guirardel, and D. Osin.  Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces.  Mem. Amer. Math. Soc., 245(1156):v+152, 2017.  doi:10.1090/memo/1156.
  * [Dix90] John Dixon.  Most finitely generated permutation groups are free.  Bull. Lond. Math. Soc., 22(3):222–226, 1990.  doi:10.1112/blms/22.3.222.
  * [FMMM18] Pierre Fima, François Le Maître, Julien Melleray, and Soyoung Moon.  Homogeneous actions on Urysohn spaces.  arXiv:1805.02477 [math], May 2018.  URL: http://arxiv.org/abs/1805.02477, arXiv:1805.02477.
  * [FMS15] Pierre Fima, Soyoung Moon, and Yves Stalder.  Highly transitive actions of groups acting on trees.  Proceedings of the American Mathematical Society, 143(12):5083–5095, August 2015.  URL: http://www.ams.org/proc/2015-143-12/S0002-9939-2015-12659-7/, doi:10.1090/proc/12659.
  * [GG13] Shelly Garion and Yair Glasner.  Highly transitive actions of $\operatorname{{Out}}({F}_{n})$.  Groups, Geometry, and Dynamics, 7(2):357–376, May 2013.  URL: https://www.ems-ph.org/journals/show_abstract.php?issn=1661-7207&vol=7&iss=2&rank=4, doi:10.4171/GGD/185.
  * [GGS20] Tsachik Gelander, Yair Glasner, and Gregory Soĭfer.  Maximal subgroups of countable groups, a survey, January 2020.  arXiv: 1909.09361.  URL: http://arxiv.org/abs/1909.09361.
  * [GM91] A.M.W. Glass and Stephen McCleary.  Highly transitive representations of free groups and free products.  Bull. Aust. Math. Soc., 43(1):19–35, 1991.  doi:10.1017/S0004972700028744.
  * [Gun92] Steven Gunhouse.  Highly transitive representations of free products on the natural numbers.  Arch. Math., 58(5):435–443, 1992.  doi:10.1007/BF01190113.
  * [Hic88] Kenneth Hickin.  Some Applications of Tree-Limits to Groups. Part 1\.  Transactions of the American Mathematical Society, 305(2):797–839, 1988.  doi:10.2307/2000887.
  * [Hic92] Kenneth Hickin.  Highly transitive Jordan representations of free products.  J. Lond. Math. Soc., II. Ser., 46(1):81–91, 1992.  doi:10.1112/jlms/s2-46.1.81.
  * [HO16] Michael Hull and Denis Osin.  Transitivity degrees of countable groups and acylindrical hyperbolicity.  Israel Journal of Mathematics, 216(1):307–353, October 2016.  URL: http://link.springer.com/10.1007/s11856-016-1411-9, doi:10.1007/s11856-016-1411-9.
  * [HP11] Pierre de la Harpe and Jean-Philippe Préaux.  C*-simple groups: Amalgamated free products, HNN extensions, and fundamental groups of 3-manifolds.  Journal of Topology and Analysis, 3(4):451–489, December 2011.  doi:10.1142/S1793525311000659.
  * [Iva20] Nikolay Ivanov.  Two Families of Examples of Groups Acting on Trees with Nontrivial Quasi-Kernels.  arXiv:2003.10196 [math], March 2020.  arXiv:2003.10196.
  * [JM13] Kate Juschenko and Nicolas Monod.  Cantor systems, piecewise translations and simple amenable groups.  Annals of Mathematics, 178(2):775–787, September 2013.  URL: http://annals.math.princeton.edu/2013/178-2/p07, doi:10.4007/annals.2013.178.2.7.
  * [Kit12] Daniel Kitroser.  Highly-transitive actions of surface groups.  Proc. Am. Math. Soc., 140(10):3365–3375, 2012.  doi:10.1090/S0002-9939-2012-11195-5.
  * [KL04] Johannes Kellendonk and Mark Lawson.  Partial actions of groups.  International Journal of Algebra and Computation, 14(01):87–114, February 2004.  URL: https://www.worldscientific.com/doi/abs/10.1142/S0218196704001657, doi:10.1142/S0218196704001657.
  * [LB16] Adrien Le Boudec.  Groups acting on trees with almost prescribed local action.  Commentarii Mathematici Helvetici, 91(2):253–293, April 2016.  URL: https://www.ems-ph.org/journals/show_abstract.php?issn=0010-2571&vol=91&iss=2&rank=3, doi:10.4171/CMH/385.
  * [LB17] Adrien Le Boudec.  C*-simplicity and the amenable radical.  Inventiones mathematicae, 209(1):159–174, July 2017.  doi:10.1007/s00222-016-0706-0.
  * [LBMB19] Adrien Le Boudec and Nicolás Matte Bon.  Triple transitivity and non-free actions in dimension one.  arXiv:1906.05744 [math], June 2019.  URL: http://arxiv.org/abs/1906.05744, arXiv:1906.05744.
  * [LS01] Roger C. Lyndon and Paul E. Schupp.  Combinatorial Group Theory.  Classics in Mathematics. Springer, Berlin, Heidelberg, reprint of the 1977 edition edition, 2001.  URL: http://link.springer.com/10.1007/978-3-642-61896-3, doi:10.1007/978-3-642-61896-3.
  * [Mal40] A.I. Malcev.  On isomorphic matrix representations of infinite groups.  Rec. Math. [Mat. Sbornik] N.S., 8 (50):405–422, 1940.
  * [Mat06] Hiroki Matui.  Some remarks on topological full groups of Cantor minimal systems.  International Journal of Mathematics, 17(02):231–251, February 2006\.  URL: https://www.worldscientific.com/doi/abs/10.1142/S0129167X06003448, doi:10.1142/S0129167X06003448.
  * [Mat18] Nicolás Matte Bon.  Rigidity properties of full groups of pseudogroups over the Cantor set.  arXiv:1801.10133 [math], January 2018.  URL: http://arxiv.org/abs/1801.10133, arXiv:1801.10133.
  * [McD77] T.P. McDonough.  A permutation representation of a free group.  Q. J. Math., Oxf. II. Ser., 28:353–356, 1977.  doi:10.1093/qmath/28.3.353.
  * [Mes72] Stephen Meskin.  Nonresidually finite one-relator groups.  Trans. Amer. Math. Soc., 164:105–114, 1972.  doi:10.2307/1995962.
  * [MO15] Ashot Minasyan and Denis Osin.  Acylindrical hyperbolicity of groups acting on trees.  Mathematische Annalen, 362(3):1055–1105, August 2015.  doi:10.1007/s00208-014-1138-z.
  * [MS13] Soyoung Moon and Yves Stalder.  Highly transitive actions of free products.  Algebraic & Geometric Topology, 13(1):589–607, 2013.  URL: https://projecteuclid.org/euclid.agt/1513715508, doi:10.2140/agt.2013.13.589.
  * [Osi16] Denis Osin.  Acylindrically hyperbolic groups.  Transactions of the American Mathematical Society, 368(2):851–888, February 2016.  URL: http://www.ams.org/tran/2016-368-02/S0002-9947-2015-06343-0/, doi:10.1090/tran/6343.
  * [Osi19] Denis Osin.  Groups acting acylindrically on hyperbolic spaces.  In Proceedings of the International Congress of Mathematicians (ICM 2018), pages 919–939, Rio de Janeiro, Brazil, May 2019\. World Scientific.  URL: https://www.worldscientific.com/doi/abs/10.1142/9789813272880_0082, doi:10.1142/9789813272880_0082.
  * [Ser80] Jean-Pierre Serre.  Trees.  Springer-Verlag, Berlin Heidelberg, 1980.  URL: https://www.springer.com/gp/book/9783540442370.