# 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.
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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 |