# groups that act freely on trees are free

###### Theorem.

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

###### Proof.

Let $\Gamma$ be a group acting freely and without inversions by graph automorphisms on a tree $T$. Since $\Gamma$ acts freely on $T$, the quotient graph $T/\Gamma$ is well-defined, and $T$ is the universal cover of $T/\Gamma$ since $T$ is contractible. Thus by faithfulness $\Gamma\cong\pi_{1}(X/\Gamma)$. 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, $\Gamma$ is free. ∎

Title groups that act freely on trees are free GroupsThatActFreelyOnTreesAreFree 2013-03-22 13:54:23 2013-03-22 13:54:23 mps (409) mps (409) 10 mps (409) Theorem msc 20F65