Suppose I have a biregular tree $T_{m, n}$ (not necessarily locally finite), with distinct cardinal numbers $m, n$, so Aut$(T_{m, n})$ acts on $T_{m, n}$ without inversion. Let $V_m$ be those vertices of $T_{m, n}$ with valency $m$, and $V_n$ those with valency $n$. As usual, for $v \in VT_{m, n}$ define $B_d(v)$ to be those vertices at distance $d$ from $v$ in $T_{m, n}$.

Suppose we are given two transitive permutation groups $M \leq S_m$ and $N \leq S_n$. If $G \leq$ Aut$(T_{m, n})$ satisfies the following:

For all $v \in V_m$, the stabilizer $G_v$ acts on $B_1(v)$ like $M$; and

For all $v \in V_n$, the stabilizer $G_v$ acts on $B_1(v)$ like $N$,

let us say that $G$ is *locally-$(M, N)$*.

Now one can conceive of groups which are maximal in Aut$(T_{m, n})$, subject to being locally-$(M, N)$. If $H$ is such a group, then $H$ will typically have lots of nice independence properties, the most important (for me) being:

- (*) For each vertex $v \in V_n$, if $C$ is a connected component of $T_{m, n} \setminus \{v\}$ then the pointwise stabilizer $H_{(C)}$ of $C$ acts on $B_1(v)$ like a point-stabilizer in $N$.

Provided $M$ and $N$ are closed (in the permutation topology) I have a sketch of a very long combinatorial method for constructing a maximal locally-$(M, N)$ group $H$ that I believe is correct, together with proofs that $H$ satisfies some nice independence properties (like property (*) I gave above). However, this idea of maximal locally-$(M, N)$ subgroups of Aut$(T_{m, n})$ seems to me to be a very natural thing to study when looking at groups acting on trees (indeed, see my point (2) below), and I do not wish to spend time laboriously constructing this group in the paper I'm currently writing unless I need to. So, my question is the following.

Question: Given two transitive closed permutation groups $M \leq S_m$ and $N \leq S_n$, is there a paper which looks at maximal subgroup(s) $H$ of the automorphism group of the biregular tree $T_{m, n}$ for which $H_v$ induces $M$ (if the valency of $v$ is $m$) or $N$ (if the valency of $v$ is $n$) on $B_1(v)$, for all $v \in VT_{m, n}$?

I should add a couple of points.

I'm aware that there are other ways to construct this group $H$ (for example using iterated wreath products to obtain point-stabilizers and an edge-stabilizer for $H$ and then forming the amalgamated free product). I'm only interested in alternative ways of constructing $H$ if the construction and the proof of the above property (*) together is particularly short and/or elegant. What I'm primarily interested in is a reference for the existence of $H$ and a reference for some of the independence properties of $H$ which I can cite instead of having to construct them myself in my paper.

I am aware of the universal group $U(F)$ (where $F$ is a finite group) from section 3.2 of Burger and Mozes' paper

*Groups acting on trees from local to global structure*. They prove that this group satisfies Tits' independence property. This is exactly the thing I am looking for, only I need it to work for non-locally-finite biregular trees.