groups that act freely on trees are free


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


Let Γ be a group acting freely and without inversions by graph automorphismsMathworldPlanetmath on a tree T. Since Γ acts freely on T, the quotient graph T/Γ is well-defined, and T is the universal cover of T/Γ since T is contractible. Thus by faithfulness Γπ1(X/Γ). Since any graph is homotopy equivalent to a wedge of circles, and the fundamental groupMathworldPlanetmathPlanetmath of such a space is free by Van Kampen’s theorem, Γ 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