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Quotient Group Well Defined Circle Group Tree Set Theoretic Graph Fundamental Group Cone Homotopy Equivalence Universal Covering Space Contractible Van Kampens Theorem Graph Homomorphism
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Groups That Act Freely On Trees Are Free
Proof.
Let 
be a group acting freely and without inversions by graph automorphisms
on a tree 
.
Since 
acts freely on 
, the quotient
graph
is well-defined, and 
is the universal cover
of 
since 
is contractible. Thus by faithfulness
. 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, 
is free.
be a group acting freely and without inversions by graph automorphisms
on a tree .
Since acts freely on , the quotient
graph
is well-defined, and is the universal cover
of since is contractible. Thus by faithfulness
. 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, is free.