Notation 2.3. Throughout the paper, when $\alpha $ is a preaction on a pointed countable set $(X, x)$ , we will denote by $\mathcal {G}_{\alpha }$ the $(m,n)$ -graph of $\alpha $ .

Proof. By the properties of the maximal forest saturation action (cf. Lemma 2.5), any vertex of $\mathcal {G}_{\beta } \smallsetminus \mathcal {G}_{\alpha }$ has $|m|$ incoming edges and $|n|$ outgoing edges if $|m|=|n|$ or if the labels of the vertices of $\mathcal {G}_{\beta }$ are infinite (i.e. the $\langle b \rangle $ -orbits of $\alpha $ are infinite). As any vertex of $\mathcal {T}_{m,n}$ also has $|m|$ incoming edges and $|n|$ outgoing edges, this tells us that $\pi $ induces a locally injective graph morphism between any connected component of $\pi ^{-1}(\widehat {\mathcal {T}})$ and $\widehat {\mathcal {T}}$ . Finally, as $\widehat {\mathcal {T}}$ is a tree, [Reference Serre13, §4.5, Lemma 5] tells us that this induced graph morphism is in fact an isomorphism.

Notation 2.9. For any preaction $\alpha $ , we denote by $p_{\alpha } : \mathrm {Sch}(\alpha ) \to \mathcal {G}_{\alpha }$ the projection.

Proof. We first build a finite set $F \subseteq \Gamma $ satisfying the following property: for every $w \in \Gamma $ , there exists $f_{w} \in F$ such that $w f_{w} \cdot v \notin \mathcal {T}^{(0)}$ .

To achieve this, using the cocompactness of $\Lambda \curvearrowright \mathcal {T}^{(0)}$ , let us fix a finite set $F_0 \subseteq \Gamma $ such that

(3.3) $$ \begin{align}\Lambda F_0 \cdot v = \mathcal{V}(\mathcal{T}^{(0)}).\end{align} $$

As $\mathcal {T}^{(0)}$ is a proper subtree of $\mathcal {T}_{m,n}$ , for any $f \in F_0$ , we also fix $\gamma _f \in \Gamma $ such that ${f\gamma _f \cdot v \notin \mathcal {T}^{(0)}}$ .

Now, let us take any $w \in \Gamma $ . If $w \cdot v \notin \mathcal {T}^{(0)}$ , then taking $f_w = 1$ implies that $w f_w \cdot v \notin \mathcal {T}^{(0)}$ . Otherwise, by (3.3), there exists some $f \in F_0$ and some $\unicode{x3bb} \in \Lambda $ such that $w \cdot v = \unicode{x3bb} f \cdot v$ . As $\mathrm {Stab}(v) = \langle b \rangle $ , there exists some $l_0 \in \mathbb {Z}$ such that $w b^{l_0} = \unicode{x3bb} f $ . Let $h_f$ be the height of $\gamma _f$ and let

$$ \begin{align*}N_f = |mn|^{h_f}.\end{align*} $$

Writing the Euclidean division of $l_0$ by $N_f$ , one gets $l_0 = - qN_f + k_0$ for some $q \in \mathbb {Z}$ and some $k_0 \in [\![ 0, N_f-1 ]\!] $ . In particular, $wb^{k_0} = \unicode{x3bb} f b^{qN_f}$ . As $h_f$ is the height of $\gamma _f$ , the defining relation of $\mathrm {BS}(m,n)$ implies that $b^{qN_f}\gamma _f \in \gamma _f \langle b \rangle $ ; thus, $wb^{k_0}\gamma _f \cdot v = \unicode{x3bb} f \gamma _f \cdot v$ . As $f \gamma _f \cdot v \notin \mathcal {T}^{(0)}$ and $\mathcal {T}^{(0)}$ is $\Lambda $ -invariant, we finally get that $wb^{k_0}\gamma _f \cdot v \notin \mathcal {T}^{(0)}$ .

So,

$$ \begin{align*}F = \{1\} \cup \bigcup_{f \in F_0}\{b^{k_0}\gamma_f \mid k_0 \in [\![ 0, N_f-1]\!]\}\end{align*} $$

is suitable.

For every $f \in F$ , using the fact that $\mathrm {Supp}(\mu )$ is a symmetric and generating set of $\Gamma $ , let us write $f = \mathfrak {g}_1^f \cdots \mathfrak {g}_{N_f}^f$ for some $N_f \in \mathbb {N}$ and $\mathfrak {g}_i^f \in \mathrm {Supp}(\mu )$ for every $i \in [\![ 1, N_f ]\!] $ . Let us denote

$$ \begin{align*}\theta = \min_{f \in F}\prod_{i=1}^{N_f}\mu(\mathfrak{g}_i^f) \in\ ]0,1[\end{align*} $$

and

$$ \begin{align*}L = \max_{f \in F}N_f.\end{align*} $$

For every $N \in \mathbb {N}^*$ and $w \in \Gamma $ , let us define the event

$$ \begin{align*}A_{N,w} := \{ w G_k \cdot v \in \mathcal{T}^{(0)} \ \text{ for all } k \in [\![ 0, NL-1 ]\!]\}. \end{align*} $$

For any $N \in \mathbb {N}$ and $w \in \Gamma $ , observe that

$$ \begin{align*} \mathbb{P}(A_{N+1,w}) &= \sum_{g \in \Gamma \mid w g \cdot v \in \mathcal{T}^{(0)}}\mathbb{P}(A_{N,w} \cap \{G_{NL-1} = g\}\text{ and } wgS_{NL} \cdots S_k \cdot v \in \mathcal{T}^{(0)} \\ &\qquad\qquad\ \ \hspace{0.8pt}\qquad \text{for all } k \in [\![ NL, (N+1)L-1 ]\!] ) \\ &= \sum_{g \in \Gamma \mid wg \cdot v \in \mathcal{T}^{(0)}}\mathbb{P}(A_{N,w} \cap \{G_{NL-1} = g\}) \cdot \mathbb{P}(wgS_{NL} \cdots S_k \cdot v \in \mathcal{T}^{(0)} \\& \qquad\qquad\ \ \hspace{0.8pt}\qquad \text{for all } k \in [\![ NL, (N+1)L-1 ]\!] ), \end{align*} $$

the last equality resulting from the independence of the $S_i$ values. Thus,

$$ \begin{align*} \mathbb{P}(A_{N+1,w}) &\leq \sum_{g \in \Gamma \mid wg \cdot v \in \mathcal{T}^{(0)}}\mathbb{P}(A_{N,w} \cap \{G_{NL-1} = g\})\cdot \mathbb{P}(S_{NL} \cdots S_{NL + N_{f_{wg}}-1} \neq f_{wg} ) \\ &\leq (1-\theta) \sum_{g \in \Gamma \mid wg \cdot v \in \mathcal{T}^{(0)}}\mathbb{P}(A_{N,w} \cap \{G_{NL-1} = g\}) \\ &= (1-\theta)\mathbb{P}(A_{N,w}). \end{align*} $$

Thus, a straightforward induction implies that $\mathbb {P}(A_{N, w}) \leq (1-\theta )^N$ , which tends to $0$ as N tends to $\infty $ .

Proof. Towards a contradiction, let us assume that, for every $f \in F_0$ , one has

(3.4) $$ \begin{align}\mathbb{P}(f G_k \cdot v \notin B_{\mathcal{T}_{m,n}}(v, M) \ \text{for all } k> 0) = 0.\end{align} $$

Let us define by induction the sequence of random variables $(\tau _i)_{i \in \mathbb {N}^*}$ as follows:

$$ \begin{align*} \tau_1 &= \inf \{k> 0 \mid G_k \cdot v \in B_{\mathcal{T}_{m,n}}(v, M)\}; \quad \text{and, for every }i \geq 1, \\ \tau_{i+1} &= \inf\{k > \tau_i \mid G_k \cdot v \in B_{\mathcal{T}_{m,n}}(v, M) \}. \end{align*} $$

For every i, notice that $\tau _i$ is a stopping time with respect to the Markov chain $(G_k)_{k \in \mathbb {N}}$ . Applying (3.4) to $f=1$ , we get that $\tau _1 < \infty $ almost surely. Assuming by induction that $\tau _i < \infty $ almost surely, we get that

$$ \begin{align*} & \mathbb{P}(\tau_{i+1} < \infty)\\ & \quad= \mathbb{P}(\text{there exists } k> \tau_i, \ G_k \cdot v \in B_{\mathcal{T}_{m,n}}(v, M) ) \\ &\quad= \sum_{f \in F_0}\mathbb{P}(\text{there exists } k > \tau_i, S_1 \cdots S_k \cdot v \in B_{\mathcal{T}_{m,n}}(v, M) \text{ and } S_1 \cdots S_{\tau_i} \in \langle b \rangle f ) \\ & \quad=\sum_{f \in F_0}\mathbb{P}(\text{there exists } k > \tau_i, f \cdot S_{\tau_i +1} \cdots S_k \in B_{\mathcal{T}_{m,n}}(v, M) \text{ and } S_1 \cdots S_{\tau_i} \in \langle b \rangle f ) \\ & \quad= \sum_{f \in F_0}\mathbb{P}(\text{there exists } k > \tau_i, f \cdot S_{\tau_i +1} \cdots S_k \in B_{\mathcal{T}_{m,n}}(v, M)) \mathbb{P}(S_1 \cdots S_{\tau_i} \in \langle b \rangle f ) \\&\quad \text{ (by independence of the }S_k\text{ values)} \\ & \quad = \sum_{f \in F_0}\mathbb{P}(S_1 \cdots S_{\tau_i} \in \langle b \rangle f ) \\ & \quad \text{ (as the }S_j\text{ values are identically distributed and using }(3.4)) \\ & \quad=1. \end{align*} $$

This proves that, almost surely, $G_k \cdot v \in B_{\mathcal {T}_{m,n}}(v, M)$ for infinitely many k values, which contradicts the fact that $(G_k \cdot v)_{k \in \mathbb {N}}$ converges almost surely to a random end.

Proof. Let us denote by $\mathcal {T}_e$ the half-tree of e. As it contains $\mathcal {T}^{(0)}$ , one has $w_0 s_1 \cdots s_{m_0} \cdot v \in \mathcal {T}_e$ . Thus, as $w_0 s_1 \cdots s_k \cdot v$ never returns in $B_{\mathcal {T}_{m,n}}(w_0 \cdot v, M)$ for $k> l_0$ , the fact that $d(w_0 s_1 \cdots s_k \cdot v, w_0 s_1 \cdots s_{k+1} \cdot v) \leq M$ for every k (by definition of M) implies that the sequence of vertices $(w_0 s_1 \cdots s_k \cdot v)_{k> l_0}$ remains in $\mathcal {T}_e$ . Let us assume that e is positive. In particular, the geodesic between $w_0 \cdot v$ and $w_0 \cdot s_1 \cdots s_k \cdot v$ begins with the positive edge e for every $k> l_0$ . Thus, the geodesic between v and $s_1 \cdots s_k \cdot v$ begins with the positive edge $w_0^{-1} \cdot e$ for every $k> l_0$ . So, by the ‘more specifically’ part of Remark 2.2, the normal form of $s_{1} \cdots s_k$ begins with $b^rt$ for some $r \in \mathbb {Z}$ and for every $k> l_0$ . Consequently, the normal form of $t^{-1}bts_{1} \cdots s_k$ begins with $t^{-1}$ , which implies (using again Remark 2.2) that the geodesic that connects v and $t^{-1}bts_{1} \cdots s_k \cdot v$ begins with a negative edge for every $k> l_0$ . Thus, the geodesic that connects $w_0 \cdot v$ and $w_0 \cdot t^{-1}bts_{1} \cdots s_k \cdot v$ begins with a negative edge; hence, $w_0 t^{-1}bts_{1} \cdots s_k \cdot v \notin \mathcal {T}_e$ so $w_0\eta _e s_{1} \cdots s_k \cdot v \notin \mathcal {T}^{(0)}$ for every $k> l_0$ . If e is negative, the same argument shows that the normal form of $tbt^{-1}s_{1} \cdots s_k$ begins with t and, consequently, that $w_0\eta _e s_{1} \cdots s_k \cdot v \notin \mathcal {T}^{(0)}$ for every $k> l_0$ .

Proof. Using the fact that $\mathrm {Supp}(\mu )$ generates $\Gamma $ as a semi-group, let us fix a writing

$$ \begin{align*}g = \mathfrak{g}_1^g \cdots \mathfrak{g}_{N_g}^g\end{align*} $$

for every $g \in F_0 \cup \{tbt^{-1},t^{-1}bt\}$ , where $\mathfrak {g}_1^g, \ldots , \mathfrak {g}_{N_g}^g \in \mathrm {Supp}(\mu )$ . Let us also define

$$ \begin{align*}q_0 = \min_{(\eta,g) \in \{tbt^{-1}, t^{-1}bt\} \times F_0}\prod_{i=1}^{N_{\eta}}\mu(\mathfrak{g}_i^{\eta}) \cdot \prod_{i=1}^{N_g}\mu(\mathfrak{g}_i^{g}).\end{align*} $$

Let

$$ \begin{align*}\kappa = \max_{(\eta, g) \in \{tbt^{-1}, t^{-1}bt\} \times F_0, (i,j) \in [\![ 1, N_{\eta}]\!] \times [\![ 1, N_g]\!]} \bigg\{\mathfrak{h}\bigg(\prod_{k=1}^i \mathfrak{g}_k^{\eta}\bigg),\mathfrak{h}\bigg(\eta \cdot \prod_{k=1}^j \mathfrak{g}_k^g\bigg), \mathfrak{h}\bigg(\prod_{k=1}^j \mathfrak{g}_k^g\bigg)\bigg\} \end{align*} $$

and let us take

$$ \begin{align*}L = \kappa + 1.\end{align*} $$

Now, let $w_0 \in \Gamma $ such that $w_0 \cdot v \notin \mathcal {N}_{\mathcal {T}_{m,n},L}(\mathcal {T}^{(0)})$ . Let us denote by e the edge with source $w_0 \cdot v$ that points towards $\mathcal {T}^{(0)}$ .

Using Lemma 3.8, let us fix some $f \in F_0$ such that

$$ \begin{align*}\mathbb{P}(f G_k \cdot v \notin B_{\mathcal{T}_{m,n}}(v, M) \ \text{for all } k> 0 ) > 0.\end{align*} $$

We denote by p this probability. Notice that, as $\Gamma $ acts on $\mathcal {T}_{m,n}$ by isometries, one also has

(3.5) $$ \begin{align}\mathbb{P}(w_0f G_k \cdot v \notin B_{\mathcal{T}_{m,n}}(w_0 \cdot v, M) \ \text{for all } k{\kern-1pt}>{\kern-1pt} 0 ) \nonumber &= \mathbb{P}(f G_k \cdot v {\kern-1pt}\notin{\kern-1pt} B_{\mathcal{T}_{m,n}}(v, M) \ \text{for all } k {\kern-1pt}>{\kern-1pt} 0 ) \\ &=p. \end{align} $$

The definition of $\kappa $ implies that, for every $(\eta ,g) \in \{tbt^{-1},t^{-1}bt\} \times F_0$ :

• • for every $i \in [\![ 1, N_g]\!] $ , the vertex $u_i := w_0 \mathfrak {g}_1^g \cdots \mathfrak {g}_i^g \cdot v$ satisfies $d(w_0 \cdot v, u_i) \leq \kappa $ ;

• • for every $i \in [\![ 1, N_g]\!] $ , the vertex $v_i :=w_0 \eta \mathfrak {g}_1^g \cdots \mathfrak {g}_i^g \cdot v$ satisfies $d(w_0 \cdot v, v_i) \leq \kappa $ ;

• • for every $i \in [\![ 1, N_{\eta }]\!] $ , the vertex $w_i :=w_0 \mathfrak {g}_1^{\eta } \cdots \mathfrak {g}_i^{\eta } \cdot v$ satisfies $d(w_0 \cdot v, w_i) \leq \kappa $ .

In particular, the triangle inequality implies that, if x is one of the $u_i$ , $v_i$ , $w_i$ values, then

$$ \begin{align*}d(\mathcal{T}^{(0)},x) &\geq d(\mathcal{T}^{(0)},w_0 \cdot v) - \kappa \\ &\geq L-\kappa \\&=1; \end{align*} $$

thus, $x \notin \mathcal {T}^{(0)}$ . Together with Lemma 3.9, this implies that

$$ \begin{align*} &\mathbb{P}(w_0 G_k \notin \mathcal{T}^{(0)}\ \text{for all } k> 0) \\ &\quad \geq \mathbb{P}(S_1 = \mathfrak{g}_1^{\eta_e}, \ldots, S_{N_{\eta_e}} = \mathfrak{g}_{N_{\eta_e}}^{\eta_e}, S_{N_{\eta_e}+1} = \mathfrak{g}_1^f, \ldots, S_{N_{\eta_e} + N_f} = \mathfrak{g}_{N_f}^f\\ & \qquad\text{and } w_0fS_{N_{\eta_e}+N_f+1}\cdots S_k \cdot v \notin B_{\mathcal{T}_{m,n}}(w_0 \cdot v, M),\ \text{for all } k > N_{\eta_e}+N_f \\ & \qquad\text{and } \text{ there exists } k > N_{\eta_e}+N_f, w_0fS_{N_{\eta_e}+N_f+1}\cdots S_k \cdot v \in \mathcal{T}^{(0)}) \\ &\quad\geq q_0 \mathbb{P}(w_0f G_k \cdot v \notin B_{\mathcal{T}_{m,n}}(w_0 \cdot v, M)\ \text{for all } k > 0 \text{ and } \text{ there exists } k > 0,\\& \qquad w_0f G_k \cdot v \in\mathcal{T}^{(0)}), \end{align*} $$

the last inequality resulting from the fact that the $S_i$ values are independent and identically distributed. Likewise, using again the previous remark:

$$ \begin{align*} &\mathbb{P}(w_0 G_k \notin \mathcal{T}^{(0)}\ \text{for all } k> 0) \\ &\quad \geq \mathbb{P}(S_1 = \mathfrak{g}_1^f, \ldots, S_{N_f} = \mathfrak{g}_{N_f}^f \text{ and } w_0fS_{N_f+1} \cdots S_k \notin \mathcal{T}^{(0)}\ \text{for all } k > N_f) \\ &\quad\geq q_0\mathbb{P}(w_0fG_k \notin \mathcal{T}^{(0)}\ \text{for all } k > 0). \end{align*} $$

Decomposing

$$ \begin{align*}p &= \mathbb{P}(w_0f G_k \cdot v \notin B_{\mathcal{T}_{m,n}}(w_0 \cdot v, M)\ \text{for all } k> 0 \text{ and } \text{ there exists } k>0,\\ & \qquad w_0f G_k \cdot v \in \mathcal{T}^{(0)})\\ &\quad +\mathbb{P}(w_0f G_k \cdot v \notin B_{\mathcal{T}_{m,n}}(w_0 \cdot v, M)\ \text{for all } k > 0 \text{ and } w_0f G_k \cdot v \notin \mathcal{T}^{(0)}\\ & \qquad \text{for all } k > 0) \end{align*} $$

we thus get

$$ \begin{align*} p &\leq \frac{2}{q_0} \mathbb{P}(w_0 G_k \cdot v \notin \mathcal{T}^{(0)}\ \text{for all } k> 0), \end{align*} $$

i.e.

$$ \begin{align*}\mathbb{P}(w_0 G_k \cdot v \notin \mathcal{T}^{(0)}\ \text{for all } k> 0) \geq \frac{pq_0}{2}.\end{align*} $$

Hence, setting $q := {pq_0}/{2}$ concludes the proof of the lemma.

Proof of Proposition 3.6.

The fact that the random walk $G_k \cdot v$ converges almost surely to a random end $\xi \in \partial \mathcal {T}_{m,n}$ results from Theorem 3.7. So, it remains to show that $\xi \notin \partial \mathcal {T}^{(0)}$ almost surely.

Let L and q be provided by Lemma 3.10. As $\mathcal {N}_{\mathcal {T}_{m,n},L}(\mathcal {T}^{(0)})$ is still a proper $\Lambda $ -invariant subtree of $\mathcal {T}_{m,n}$ on which $\Lambda $ acts cocompactly, Lemma 3.4 tells us that

(3.6) $$ \begin{align}\mathbb{P}(\text{there exists } k \geq 0, wG_k \cdot v \notin \mathcal{N}_{\mathcal{T}_{m,n}, L}(\mathcal{T}^{(0)}) ) = 1 \text{ for every }w \in \Gamma. \end{align} $$

Let us define inductively the random variables $\tau _i, \sigma _i$ that are the (possibly infinite) indices of the ith exit from $\mathcal {N}_{\mathcal {T}_{m,n},L}(\mathcal {T}^{(0)})$ (respectively return to $\mathcal {T}^{(0)}$ ):

$$ \begin{align*} \tau_1 &= \min \{k> 0, G_k \cdot v \notin \mathcal{N}_{\mathcal{T}_{m,n},L}(\mathcal{T}^{(0)})\}; \\ \sigma_i &= \min\{k > \tau_i, G_k \cdot v \in \mathcal{T}^{(0)}\};\\ \tau_{i+1} &= \min\{k > \sigma_i, G_k \cdot v \notin \mathcal{N}_{\mathcal{T}_{m,n},L}(\mathcal{T}^{(0)})\}. \end{align*} $$

Let us define the event $\Sigma _i := \{\sigma _i < \infty \}$ . Equation (3.6) implies that

$$ \begin{align*}\{G_k \cdot v \in \mathcal{T}^{(0)} \text{ for infinitely many }k\text{'s} \} = \bigcap_{i \in \mathbb{N}^*}\Sigma_i; \end{align*} $$

thus, we want to prove that $\mathbb {P}(\bigcap _{i \in \mathbb {N}^*}\Sigma _i)=0$ . As $\Sigma _{i+1} \subseteq \Sigma _i$ for every $i \in \mathbb {N}^*$ , one has $\mathbb {P}(\bigcap _{i \in \mathbb {N}^*} \Sigma _i) = \lim _{i \to +\infty }\mathbb {P}(\Sigma _i)$ . For every $i \in \mathbb {N}$ :

$$ \begin{align*} \mathbb{P}(\Sigma_{i+1}) &= \mathbb{P}(\tau_{i+1} < \infty \text{ and } \sigma_{i+1} < \infty)\\ &= \sum_{w_0 \in \Gamma, w_0 \cdot v \notin \mathcal{N}_{\mathcal{T}_{m,n}, L}(\mathcal{T}^{(0)})}\mathbb{P}(\tau_{i+1} < \infty \text{ and } S_1 \cdots S_{\tau_{i+1}} = w_0 \\ &\qquad\qquad\hspace{1pt}\qquad\qquad\qquad \text{and } \text{ there exists } \sigma> \tau_{i+1}, S_1 \cdots S_{\sigma} \cdot v \in \mathcal{T}^{(0)} )\\ &= \sum_{w_0 \in \Gamma, w_0 \cdot v \notin \mathcal{N}_{\mathcal{T}_{m,n}, L}(\mathcal{T}^{(0)})}\mathbb{P}(\tau_{i+1} < \infty \text{ and } S_1 \cdots S_{\tau_{i+1}} = w_0 \\ &\qquad\qquad\hspace{1pt}\qquad\qquad\qquad \text{and } \text{ there exists } \sigma> \tau_{i+1}, w_0 S_{\tau_{i+1}+1} \cdots S_{\sigma} \cdot v \in \mathcal{T}^{(0)} )\\ &= \sum_{w_0 \in \Gamma, w_0 \cdot v \notin \mathcal{N}_{\mathcal{T}_{m,n}, L}(\mathcal{T}^{(0)})}\mathbb{P}(\tau_{i+1} < \infty \text{ and } S_1 \cdots S_{\tau_{i+1}} = w_0) \\ & \qquad\qquad\hspace{-0.5pt}\qquad\qquad\qquad \times \mathbb{P}( \text{there exists } \sigma> \tau_{i+1}, w_0 S_{\tau_{i+1}+1} \cdots S_{\sigma} \cdot v \in \mathcal{T}^{(0)} ), \end{align*} $$

the last equality resulting from the independence of the $S_k$ values and the fact that $\tau _{i+1}$ is a stopping time with respect to the random walk with step distribution $(S_k)_{k \in \mathbb {N}}$ . As the $S_k$ values are identically distributed, we thus get

$$ \begin{align*} \mathbb{P}(\Sigma_{i+1}) &= \sum_{w_0 \in \Gamma, w_0 \cdot v \notin \mathcal{N}_{\mathcal{T}_{m,n}, L}(\mathcal{T}^{(0)})}\mathbb{P}(\tau_{i+1} < \infty \text{ and } S_1 \cdots S_{\tau_{i+1}} = w_0)\\ & \qquad\qquad\qquad\hspace{-0.5pt}\qquad\qquad \times \mathbb{P}( \text{there exists } \sigma'> 0, w_0 S_{1} \cdots S_{\sigma'} \cdot v \in \mathcal{T}^{(0)} ) \\ &\leq (1-q)\sum_{w_0 \in \Gamma, w_0 \cdot v \notin \mathcal{N}_{\mathcal{T}_{m,n}, L}(\mathcal{T}^{(0)})}\mathbb{P}(\tau_{i+1} < \infty \text{ and } S_1 \cdots S_{\tau_{i+1}} = w_0)\\ &\text{ (by Lemma~}3.10) \\ &= (1-q)\mathbb{P}(\tau_{i+1} < \infty)\\ &\leq (1-q)\mathbb{P}(\Sigma_i). \end{align*} $$

A straightforward induction thus implies that $\mathbb {P}(\Sigma _i) \leq (1-q)^{i-1}$ for every $i \in \mathbb {N}^*$ , which proves that $\mathbb {P}(\bigcap _{i \in \mathbb {N}^*}\Sigma _i) = 0$ .

Proof. Let $\Lambda = \mathrm {Stab}_{\alpha }(x)$ and let $\pi : \mathcal {T}_{m,n} \to \Lambda \backslash \mathcal {T}_{m,n} = \mathcal {G}_{\beta }$ be the projection. Let $\mathcal {T}^{(0)} = \pi ^{-1}(\mathcal {G}_{\alpha })$ . By Remark 2.4, $\mathcal {T}^{(0)}$ is a proper $\Lambda $ -invariant subtree of $\mathcal {T}_{m,n}$ on which $\Lambda $ acts cocompactly (the quotient graph $\Lambda \backslash \mathcal {T}^{(0)}$ being $\mathcal {G}_{\alpha }$ ). So by Proposition 3.6, the sequence $(G_k \cdot v)_{k \in \mathbb {N}}$ converges almost surely to a random end $\xi \in \partial \mathcal {T}_{m,n} \smallsetminus \partial \mathcal {T}^{(0)}$ . Let $e \in \mathcal {E}(\mathcal {T}_{m,n})$ such that $\mathbf {s}(e) \in \mathcal {T}^{(0)}$ , $\mathbf {t}(e) \notin \mathcal {T}^{(0)}$ and $\xi $ belongs to the half-tree $\mathcal {T}_e$ of e. For k large enough, the sequence $G_k \cdot v$ remains in $\mathcal {T}_e$ . Furthermore, $\pi $ induces a graph isomorphism between $\mathcal {T}_e$ and $\pi (\mathcal {T}_e)$ by Proposition 2.8 applied to $f = \pi (e)$ . Thus, the sequence

$$ \begin{align*} p_{\beta}(x \cdot G_k) &= (\Lambda G_k)\langle b \rangle \\ &= \Lambda (G_k \langle b \rangle) \\ &=\pi(G_k \cdot v) \end{align*} $$

converges almost surely to the random end $\pi (\xi )$ in $\pi (\mathcal {T}_e)$ .

Proof. Let us define the subset

$$ \begin{align*}X_1' = X_1 \cup \{p_{\beta_1}^{-1}(p_{\beta_1}(x_1 \cdot w)) \mid w\text{ subword of }g_1\}\end{align*} $$

of $Y_1$ and the subset

$$ \begin{align*}X_2' = X_2 \cup \{p_{\beta_2}^{-1}(p_{\beta_2}(x_2 \cdot w)) \mid w\text{ subword of }g_3^{-1}\}\end{align*} $$

of $Y_2$ . For $i \in \{1,2\}$ , let us denote by $\beta _i'$ the restriction of $\beta _i$ defined on $X_i'$ .

As $P = \infty $ or $|m|=|n|$ , the following arguments will take place in a half-tree of $\mathcal {G}_{\alpha _i}^c$ (for $i \in \{1,2\}$ ), which is isomorphic to a half-tree of $\mathcal {T}_{m,n}$ by Proposition 2.8; in particular, an edge path of $\mathcal {G}_{\alpha _i}^c$ that derives from a vertex of $\mathcal {G}_{\alpha _i}^c$ and a reduced word $\gamma $ is reduced, and its length is the height $\mathfrak {h}(\gamma )$ of $\gamma $ .

Especially, the edge path deriving from $(g_2,x_1 \cdot g_1)$ is reduced in $\mathcal {G}_{\alpha _1}^c$ . Thus, one can write the normal form of $g_2$ as $g_2 = uv$ , where all the subwords of u are defined on $x_1 \cdot g_1$ and no non-empty subword of v is defined on $x_1 \cdot g_1u$ (for the preaction $\beta _1'$ ). One has

$$ \begin{align*} \mathfrak{h}(u)&= d_{\mathcal{G}_{\beta_1}}(p_1(x_1 \cdot g_1),p_1(x_1 \cdot g_1u)) \\ &\leq d_{\mathcal{G}_{\beta_1}}(p_1(x_1 \cdot g_1),\mathcal{G}_{\alpha_1}). \end{align*} $$

Likewise, one can write the reduced normal form of $g_2$ as $g_2 = u'v'$ , where all subwords of $v^{\prime -1}$ are defined on $x_2 \cdot g_3^{-1}$ and no non-empty subword of $u^{\prime -1}$ is defined on $x_2 \cdot g_3^{-1} v^{\prime -1}$ (for the preaction $\beta _2')$ . One has

$$ \begin{align*} \mathfrak{h}(v')&= d_{\mathcal{G}_{\beta_2}}(p_{\beta_2}(x_2 \cdot g_3^{-1}),p_{\beta_2}(x_2 \cdot g_3^{-1}v^{\prime -1})) \\ &\leq d_{\mathcal{G}_{\beta_2}}(p_{\beta_2}(x_2 \cdot g_3^{-1}), \mathcal{G}_{\alpha_2}). \end{align*} $$

Thus, the third assumption implies that

$$ \begin{align*}\mathfrak{h}(g_2)&=d_{\mathcal{G}_{\beta_1}}(p_{\beta_1}(x_1 \cdot g_1),p_{\beta_1}(x_1 \cdot g_1g_2))\\& \geq \mathfrak{h}(u)+\mathfrak{h}(v')+2,\end{align*} $$

so the initial subword u of $g_2$ is in fact an initial subword of $u'$ , and one can write the normal form of $g_2$ as $g_2 = u u" v'$ , where $\mathfrak {h}(u") \geq 2$ .

Let us write $u" = t^{\varepsilon }\mathfrak {m}t^{\eta }$ for some reduced word $\mathfrak {m}$ , where:

• • $\varepsilon , \eta \in \{1,-1\}$ ;

• • $t^{\varepsilon }$ is not defined on $x_1 \cdot g_1u$ (for the preaction $\beta _1'$ );

• • $t^{-\eta }$ is not defined on $x_2 \cdot g_3^{-1}v^{\prime -1}$ (for the preaction $\beta _2'$ ).

Let us denote by $b^{n_1}t^{\varepsilon _1}\cdots b^{n_r}t^{\varepsilon _r}b^{n_{r+1}}$ the reduced form of $\mathfrak {m}$ . By induction on r, we build a preaction $\gamma $ defined on a countable set S with the following properties.

• • Its $(m,n)$ -graph is an edge path $e_1,\ldots ,e_{r}$ , such that the orientation of the edge $e_i$ is the sign of $\varepsilon _i$ .

• • Its $\langle b \rangle $ -orbits share the same cardinal C, which is the common label of the vertices of $\mathcal {G}_{\alpha _i}^c$ for $i\in \{1,2\}$ by Remark 2.17 (and which is infinite if P is).

• • There exist $y_1,y_2 \in S$ such that:

  1. – $t^{-\varepsilon }$ is not defined on $y_1$ ;

  2. – $t^{\eta }$ is not defined on $y_2$ ;

  3. – $y_2 = y_1 \cdot \mathfrak {m}$ .

Finally, we merge the preactions $\beta _1'$ , $\gamma $ , and $\beta _2'$ into a single preaction defined on ${X_1' \sqcup S \sqcup X_2'}$ by defining:

• • $x_1 \cdot g_1u \cdot t^{\varepsilon } = y_1$ ;

• • $y_2 \cdot t^{\eta } = x_2 \cdot g_3^{-1}v^{\prime -1}$ .

An illustration of this construction on the level of $(m,n)$ -graphs is provided in Figure 2.

Figure 2

An illustration of the proof of Lemma 3.12.

Figure 2 Long description

There are two balls G alpha 1 and G alpha 2, each of those containing a vertex x 1 and x 2, respectively. There is a path labeled g 1 (resp. g 2) that lies in G alpha 1 (resp. that lies in a neighbourhood of G alpha 2) before going outside and reaching a vertex x 1 cdot g 1 (resp. x 2 cdot g 3 -1). Another path labeled u (resp. v' -1) goes backwards (following a portion of the previous path in the other direction), going from x 1 cdot g 1 (resp. x 2 cdot g 3 -1) to a vertex x 1 cdot g 1 u (resp. x 2 cdot g 3 -1 v' -1) which is sill outside G alpha 1 (resp. G {alpha 2}). Finally, a dashed edge path labeled u'' connects x 1 cdot g 1 u to x 2 cdot g 3 -1 v' -1.

For this new preaction $\delta $ , we get

$$ \begin{align*} x_2 \cdot g_3^{-1}v^{\prime -1} &= y_2 \cdot t^{\eta} \\ &= y_1 \cdot \mathfrak{m} \cdot t^{\eta} \\ &= x_1 \cdot g_1u t^{\varepsilon}\mathfrak{m} \cdot t^{\eta} \\ &= x_1 \cdot g_1u u", \end{align*} $$

which implies that

$$ \begin{align*} x_2 &= x_1 \cdot g_1u u"v'g_3 \\ &= x_1 \cdot g_1g_2g_3 \end{align*} $$

and $\delta $ extends both $\alpha _1'$ and $\alpha _2'$ . The $(m,n)$ -graph of $\delta $ consists of:

• • the $(m,n)$ -graph of $\beta _1'$ ;

• • the $(m,n)$ -graph of $\beta _2'$ ;

• • the edge path $e_1,\ldots ,e_r$ (with $\mathbf {s}(e_1) \in \mathcal {G}_{\beta _1'}$ and $\mathbf {t}(e_N) \in \mathcal {G}_{\beta _2'}$ ), all of whose vertices are labeled C.

Thus, as $|m|, |n|, r \geq 2$ , the vertex $\mathbf {t}(e_1)$ is not saturated. Hence, the maximal forest saturation action $\alpha $ of $\delta $ given by Lemma 2.5 has an infinite $(m,n)$ -graph, and thus satisfies the required conditions.

Proof of Theorem 1.2.

Let $\Lambda _1$ and $\Lambda _2$ be two subgroups of $\Gamma $ whose $(m,n)$ -graphs are infinite. Let us assume that:

• • either $\boldsymbol {\mathrm {Ph}}_{m,n}(\Lambda _i) = \infty $ for $i \in \{1,2\}$ ; or

• • $|m|=|n|$ and $\boldsymbol {\mathrm {Ph}}_{m,n}(\Lambda _1) = \boldsymbol {\mathrm {Ph}}_{m,n}(\Lambda _2)$ ,

and let us denote by P the common phenotype of $\Lambda _1$ and $\Lambda _2$ .

For $i \in \{1,2\}$ , let us denote by $\gamma _i$ the associated transitive right action on a pointed countable set $(X_i, x_i)$ . Let us fix $R>0$ and let $\mathcal {F}_i$ be the R-ball of $\mathcal {G}_{\gamma _i}$ around $p_{\gamma _i}(x_i)$ . We denote

$$ \begin{align*}U_i^{(R)} = \{&\gamma \text{ transitive and saturated preaction on a pointed countable set } (W_{\gamma},w_{\gamma}) \mid \\& \mathrm{Stab}_{\gamma}(w_{\gamma}) \in \mathcal{K}(\Gamma) \text{ and } (p_{\gamma}^{-1}(B_{\mathcal{G}_{\gamma}}(p_{\gamma}(w_{\gamma}),R)), w_{\gamma}) \simeq (p_{\gamma_i}^{-1}(\mathcal{F}_i), x_i) \\& \text{ (as pointed labeled graphs)}\}.\end{align*} $$

Identifying subgroups with pointed, transitive, and saturated preactions, we recall that the sets $(U_i^{(R)})_{R> 0}$ form a basis of neighborhoods of $\Lambda _i$ for the topology $\mathcal {T}_{sat}$ on the set of pointed transitive actions defined in §2.2.3 induced on $\mathcal {K}(\Gamma )$ . Moreover, $U_i^{(R)}$ is included in $\boldsymbol {\mathrm {Ph}_{m,n}}^{-1}(P)$ .

Let $\alpha _i$ be the subpreaction of $\gamma _i$ defined on a subset $X_i' \subseteq X_i$ containing $x_i$ and whose $(m,n)$ -graph is $\mathcal {F}_i$ . For $i \in \{1,2\}$ , let $\beta _i$ be the maximal forest saturation action of $\alpha _i$ given by Lemma 2.5 (defined on a countable set $Y_i$ that contains $X_i'$ ).

Let us fix $\varepsilon> 0$ . Using the fact that $\mathrm {Supp}(\mu )$ is bounded, let us denote $M := \max _{\gamma \in \mathrm {Supp}(\mu )}\mathfrak {h}(\gamma )$ . By Corollary 3.11 applied to $\beta _1$ and a sequence $(S_k)_{k \in \mathbb {N}}$ of independently $\mu $ -distributed random variables on the one hand, and to $\beta _2$ and a sequence $(S^{\prime }_k)_{k \in \mathbb {N}}$ of independently $\mu ^{-1}$ -distributed random variables (legit by Remark 2.12), denoting by $(G_k)_{k \in \mathbb {N}} = (S_1 \cdots S_k)_{k \in \mathbb {N}}$ and $(G_k')_{k \in \mathbb {N}} = (S_1' \cdots S_k')_{k \in \mathbb {N}}$ , we get that:

• • there exists some $k_1 \in \mathbb {N}$ such that

  (3.7) $$ \begin{align}\mathbb{P}(p_{\beta_1}(x_1 \cdot G_k) \notin \mathcal{N}_{\mathcal{G}_{\beta_1}, M}(\mathcal{F}_1) \ \text{for all } k \geq k_1) \geq 1 - \varepsilon;\end{align} $$

• • there exists some $k_2 \in \mathbb {N}$ such that

  (3.8) $$ \begin{align}\mathbb{P}(p_{\beta_2}(x_2 \cdot G_k') \notin \mathcal{N}_{\mathcal{G}_{\beta_2}, M}(\mathcal{F}_2) \ \text{for all } k \geq k_2) \geq 1 - \varepsilon. \end{align} $$

The choice of M implies that

(3.9) $$ \begin{align}&\mathbb{P}(p_{\beta_1}(x_1 \cdot G_{k_1}U) \notin \mathcal{F}_{1} \text{ \ for every subword }U\text{ of }S_{k_1+1}\cdots S_k\text{ and every }k \geq k_1) \nonumber \\& \quad \geq \mathbb{P}(p_{\beta_1}(x_1 \cdot G_k) \notin \mathcal{N}_{\mathcal{G}_{\beta_1}, M}(\mathcal{F}_1) \ \text{for all } k \geq k_1) \nonumber \\ &\quad \geq 1- \varepsilon \text{ by }(3.7)\end{align} $$

and

(3.10) $$ \begin{align}&\mathbb{P}(p_{\beta_2}(x_2 \cdot G^{\prime}_{k_2}U) \notin \mathcal{F}_{2} \text{ \ for every subword }U\text{ of }S_{k_2+1}'\cdots S_k'\text{ and every }k \geq k_2) \nonumber \\& \quad\geq \mathbb{P}(p_{\beta_2}(x_2 \cdot G_k') \notin \mathcal{N}_{\mathcal{G}_{\beta_2}, M}(\mathcal{F}_2) \ \text{for all } k \geq k_2) \nonumber \\ & \quad \geq 1-\varepsilon \text{ by }(3.8).\end{align} $$

We draw the attention of the reader to the fact that $S_i, S_i'$ values may not be one of the standard generators $b,b^{-1},t,t^{-1}$ , and that the term subword has to be understood in the sense of §2.2.2, i.e. with respect to the standard generators.

Let us fix some $M'$ such that, for $i \in \{1,2\}$ ,

$$ \begin{align*}M'> \max_{(h_1,\ldots ,h_{k_i}) \in \mathrm{Supp}(\mu)^{k_i}}\mathfrak{h}(h_1\cdots h_{k_i}).\end{align*} $$

For $i \in \{1,2\}$ , let us also define

$$ \begin{align*}C_i = \max_{(h_1,\ldots, h_{k_i}) \in \mathrm{Supp}(\mu)^{k_i}}d_{\mathcal{G}_{\beta_i}}(p_{\beta_i}(x_i \cdot h_1\cdots h_{k_i}), \mathcal{F}_i).\end{align*} $$

By Corollary 3.11, there exists $k_0> k_1+k_2$ such that, for any $k \geq k_0$ ,

(3.11) $$ \begin{align} \mathbb{P}(d_{\mathcal{G}_{\beta_1}}(p_1(x_1), p_1(x_1 \cdot S_1\cdots S_{k-k_2})) \geq C_1+C_2+2M'+2) \geq 1 - \varepsilon. \end{align} $$

Now, we fix $k \geq k_0$ and, for every $i \in [\![ 1, k ]\!] $ , we let $S_i' = S_{k-i+1}^{-1}$ . Let us introduce the events

$$ \begin{align*} A & := \{p_{\beta_1}(x_1 \cdot S_1\cdots S_{k_1}U) \notin \mathcal{F}_{1} \text{ \ for every subword }U\text{ of }S_{k_1+1}\cdots S_k\};\\ B & := \{p_{\beta_2}(x_2 \cdot S_1'\cdots S_{k_2}'U) \notin \mathcal{F}_{2} \text{ \ for every subword }U\text{ of }S_{k_2+1}'\cdots S_k'\};\\ C & := \{d_{\mathcal{G}_{\beta_1}}(p_1(x_1), p_1(x_1 \cdot S_1\cdots S_{k-k_2})) \geq C_1+C_2+2M'+2\}.\end{align*} $$

By (3.9), (3.10), and (3.11), we get

$$ \begin{align*} \mathbb{P}(A \cap B \cap C) \geq 1-3\varepsilon. \end{align*} $$

Let $(s_i)_{i \in \mathbb {N}} \in \mathrm {Supp}(\mu )^{\mathbb {N}}$ satisfying these three conditions, i.e.:

• • $p_{\beta _1}(x_1 \cdot s_1\cdots s_{k_1}u) \notin \mathcal {F}_{1}$ for every subword u of $s_{k_1+1}\cdots s_k$ ;

• • $p_{\beta _2}(x_2 \cdot s_k^{-1}\cdots s_{k-k_2+1}^{-1}v) \notin \mathcal {F}_{2}$ for every subword v of $s_{k-k_2}^{-1}\cdots s_1^{-1}$ ;

• • $d_{\mathcal {G}_{\beta _1}}(p_{\beta _1}(x_1), p_{\beta _1}(x_1 \cdot s_1\cdots s_{k-k_2})) \geq C_1+C_2+2M'+2$ .

Then,

$$ \begin{align*} &d_{\mathcal{G}_{\beta_1}}(p_{\beta_1}(x_1 \cdot s_1\cdots s_{k_1}), p_{\beta_1}(x_1 \cdot s_1\cdots s_{k_1} \cdot s_{k_1+1} \cdots s_{k-k_2})) \\& \quad \geq d_{\mathcal{G}_{\beta_1}}(p_{\beta_1}(x_1),p_{\beta_1}(x_1 \cdot s_1\cdots s_k)) - d_{\mathcal{G}_{\beta_1}}(p_{\beta_1}(x_1), p_{\beta_1}(x_1 \cdot s_1\cdots s_{k_1})) \\ & \qquad - d_{\mathcal{G}_{\beta_1}}(p_{\beta_1}(x_1\cdot s_1\cdots s_{k-k_2}), p_{\beta_1}(x_1 \cdot s_1\cdots s_k)) \text{\ (by the triangle inequality)}\\ &\quad\geq (C_1+C_2+2M'+2) - M' - M' \\ &\quad= C_1+C_2+2 \\ &\quad\geq d_{\mathcal{G}_{\beta_1}}(\mathcal{F}_1,p_{\beta_1}(x_1 \cdot s_1 \cdots s_{k_1})) +d_{\mathcal{G}_{\beta_2}}(\mathcal{F}_2,p_{\beta_2}(x_2 \cdot s_{k}^{-1} \cdots s_{k-k_2+1}^{-1}))+2, \end{align*} $$

which implies that the preactions $\alpha _1, \alpha _2$ and the reduced forms $g_1,g_2,g_3$ of the three elements $s_1 \cdots s_{k_1}$ , $s_{k_1+1} \cdots s_{k-k_2}$ , and $s_{k-k_2+1} \cdots s_k$ of $\Gamma $ satisfy the assumptions of Lemma 3.12. Thus, by Lemma 3.12, there exists a saturated preaction $\alpha $ defined on a pointed countable set $(X,x_1)$ that contains $X_1'$ and $X_2'$ as disjoint subsets, whose $(m,n)$ -graph is infinite, and such that:

• • $\alpha $ extends both $\alpha _1'$ and $\alpha _2'$ ;

• • $x_1 \cdot s_1\cdots s_k = x_2$ .

We proved that for every $R>0$ and every $\varepsilon> 0$ , there exists $k_0 \in \mathbb {N}$ such that, for every $k \geq k_0$ , one has

$$ \begin{align*}\mathbb{P}(\text{there exists } \Lambda \in \mathcal{K}(\Gamma)\cap \boldsymbol{\mathrm{Ph}}_{m,n}^{-1}(P): \Lambda \in U_1^{(R)} \cap G_k^{-1}U_2^{(R)}G_k) \geq 1-3\varepsilon.\end{align*} $$

Thus, as the sets $(U_i^{(R)})_{R> 0}$ form a basis of neighborhoods of $\Lambda _i$ for $\mathcal {T}_{sat}$ , which is finer than the Chabauty topology (cf. Remark 2.10), the conjugation action is topologically $\mu $ -mixing on $\mathcal {K}(\Gamma ) \cap \boldsymbol {\mathrm {Ph}_{m,n}}^{-1}(P)$ .