groups that act freely on trees are free
Groups that act freely and without inversions on trees are free.
Let be a group acting freely and without inversions by graph automorphisms on a tree . Since acts freely on , the quotient graph is well-defined, and is the universal cover of since is contractible. Thus by faithfulness . Since any graph is homotopy equivalent to a wedge of circles, and the fundamental group of such a space is free by Van Kampen’s theorem, is free. ∎