TITLE: Recognising group actions on trees from the boundary
QUESTION [9 upvotes]: Let $G$ be a group acting on a locally finite tree $T$.  Then the boundary $\partial T$ is a Cantor set on which $G$ acts by homeomorphisms (indeed by quasi-isometries under a suitable metric).  However, even if $G$ is the full automorphism group of $T$, we can't get the full quasi-isometry group of $\partial T$, as the tree structure puts further restrictions on the action.
What ways are known of identifying those group actions by homeomorphisms on the Cantor set which arise as actions on the boundary of a locally finite tree?  Is there a nice algebraic description of $\mathrm{Aut}(T)$ as a subgroup of $\mathrm{Homeo}(\partial T)$?
Edit: Here is an example of a sufficient criterion to show what I mean.  Suppose that $G$ acts on the Cantor set $X$ by homeomorphisms, with finite orbits on the clopen subsets.  Then one can produce a tree $T$ such that $G$ acts on $T$ with a global fixed point and $\partial T$ is $G$-homeomorphic to $X$.  So an action with finite orbits on the clopen subsets is automatically 'arboreal' (we do not care about which tree appears, just that it exists).

REPLY [5 votes]: I'll focus on potential topological characterizations of the action. As I mentioned in the comments above, every element of $G$ will either lie in a maximal compact subgroup of $Aut(\partial{T})$, or will have north-south dynamics (a hyperbolic element). 
If the group $G$ acts discretely, cocompactly on $\mathcal{T}$, then there's Bowditch's characterization that this is equivalent to $G$ acting properly dicontinuously  and cocompactly on the triple points in $\partial{T}$. In fact, in this case $G$ will be a virtually free hyperbolic group, and a (finite) graph of finite groups. 
I believe that $Aut(\mathcal{T})$ is closed in $Aut(\partial{\mathcal{T}})$, with the induced topology. In this case, one may assume that $G\leq Aut(\mathcal{T})$ is closed, otherwise taking its closure $\overline{G}$ in $Aut(\partial{\mathcal{T}})$ will correspond to taking its closure in $Aut(\mathcal{T})$, and if $\overline{G}\leq  Aut(\mathcal{T})$, then $G\leq Aut(\mathcal{T})$. So assume now that $G$ is closed in $Aut(\partial{\mathcal{T}})$. 
If $G$ is a compact subgroup of $Aut(\partial{\mathcal{T}})$, then I think it should
be a profinite group, which acts on a tree (elliptically with a global fixed point). 
However, I'm not sure exactly how to tell if the action of $G$ on $\partial{\mathcal{T}}$
corresponds to this group action. Let's assume from now on that $G$ is not compact. 
If $G$ does not act cocompactly on $\mathcal{T}$, then consider the limit set $\Lambda(G)\subset \partial{\mathcal{T}}$, which is the set of accumulation points of $Gx\subset \partial{\mathcal{T}}$ for any $x\in\partial{\mathcal{T}}$. One may realize this as the closure of the limit points of
hyperbolic elements of $G$. Then $G$ should act cocompactly on the convex hull   of $\Lambda(G)=\mathcal{R}$ inside of $\mathcal{T}$. So one may replace $\mathcal{T}$ with $\mathcal{R}$, and  $\partial{\mathcal{T}}$ with $\partial{\mathcal{R}}=\Lambda(G)$.  
So now assume that $G$ is a closed noncompact subgroup of $Aut(\partial{\mathcal{T}})$, and $\Lambda(G)=\partial{\mathcal{T}}$. 
Now, I believe that there should be a generalization of Bowditch's theorem. $G$ should be hyperbolic as a topological group, generated by a compact subset. I think this should be equivalent to $G$ acting properly cocompactly on the triple point space of $\partial{\mathcal{T}}$, but I don't know if this is proven (or even correct). Compare to the action of $Isom(\mathbb{H}^n)$ on $(\partial\mathbb{H}^n)^3-\Delta$, which is proper and cocompact. 
There is another possible topological characterization in terms of actions on spaces of walls, but this is really just an encoding of the tree on $\partial{T}$ in terms of how each edge partitions $\partial{T}$ into pairs of clopen sets. If $G$ is closed, but noncompact and nondiscrete, then one might be able to encode the tree by the maximal compact subgroups of $G$ and their intersections, essentially Bass-Serre theory. But I haven't thought this through.