groups that act freely on trees are free

Theorem.

Groups that act freely and without inversions on trees are free.

Proof.

Let $\mathrm{\Gamma }$ be a group acting freely and without inversions by graph automorphisms on a tree $T$.Since $\mathrm{\Gamma }$ acts freely on $T$, the quotient graph $T/\mathrm{\Gamma }$ is well-defined, and $T$ is the universal cover of $T/\mathrm{\Gamma }$ since $T$ is contractible. Thus by faithfulness $\mathrm{\Gamma }\cong {\pi }_1(X/\mathrm{\Gamma })$. 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, $\mathrm{\Gamma }$ is free.∎