I'm currently studying representation theory of groups related with trees. One of the sources I'm reading talks about "$PGL(2,\mathcal{F})$ thought of as a group of automorhpisms of its canonical tree".
By the context I could infer the following: $PGL(2,\mathcal{F})$ is supposed to be a proper closed subgroup of the automorphism group of this "canonical" tree.
I'd like to know what this tree is and what if the (whole) automorphism group of this tree is known / what it is.
Thanks!
One can make a matrix group $GL(2,\mathcal F)$ out of any commutative ring, by taking $2 \times 2$ matrices possessing a unique inverse. One can mod out by diagonal matrices to give the quotient group $PGL(2,\mathcal F)$. However, there is not always a tree associated to this construction, although there is usually some other kind of geometric object.
For instance, when $\mathcal F = \mathbb R$ then the group $PGL(2,\mathbb R)$ acts on the upper half plane with the hyperbolic metric.
On the other hand, when $\mathcal F = \mathbb Z$ then $PGL(2,\mathbb Z)$ acts very naturally on a tree.
And when $\mathcal F$ is the field of p-adic numbers $\mathbb Q_p$ then $PGL(2,\mathbb Q_p)$ acts very naturally on a tree.
I do believe there is a general theory explaining when a choice of the ring $\mathcal F$ leads to an action of $PGL(2,\mathcal F)$ on a tree, and it might seem the book you are reading would say that theory is, if one looked in the right place. But I'm not familiar with that book.
The trees can be explained in the various different cases, and a real answer to your question would contains some kind of explanation, but I'll stop with this for now.