Group action on tree

Question

Looking for different proofs of the following:

Answer 1

The vertices of $T$ are the disjoint union of cosets of $A$ and $B$ (indeed a tree on wich $G$ acts in this way is unique up to isomorphism) and you probably know how the action look like.

Your statement holds directly by definition of this action. Write down what it means that $g$ fixes a vertex $v$ and it's done.

Answer 2

Assuming you are asking for references, I would suggest the paper by Scott and Wall "Topological methods in group theory" for a topological proof, and the book by Serre "Trees" for an algebraic proof.