# Groups acting on non-locally-finite trees with independence and specified local actions

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:

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

2. 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.

1. 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.

2. 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.

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A few questions. Are $m,n$, cardinal numbers? How is the intersection $A := M \cap N$ defined when $M,N$ live in different groups? You speak about "the largest subgroup of Aut$(T_{m, n})$ to be locally-$(M, N)$", but that language presupposes uniqueness of such subgroup; is uniqueness known to be true in this context? –  Lee Mosher Sep 1 '13 at 16:19
I've edited the post following your questions Lee; hopefully it is clearer now. –  Simon Smith Sep 2 '13 at 5:53