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. ∎

Title	groups that act freely on trees are free
Canonical name	GroupsThatActFreelyOnTreesAreFree
Date of creation	2013-03-22 13:54:23
Last modified on	2013-03-22 13:54:23
Owner	mps (409)
Last modified by	mps (409)
Numerical id	10
Author	mps (409)
Entry type	Theorem
Classification	msc 20F65