CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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A group acting freely without inversions on a tree is free

Statement

If a group $G$ acts freely and without inversions on a simplicial tree, then $G$ is a free group.

Facts & Assumptions

Given: A group $G$ acting freely and without inversions on a simplicial tree $T$ .

[L1]

Bass-Serre structure identifies $G$ with the fundamental group of the quotient graph of stabilizers. (Bass-Serre structure theorem)

[L2]

A free group on a set is characterized by the universal property recorded in Free group on a set of generators.

[L3]

The relative fundamental group is obtained from the path group by killing the maximal-tree edges. (The fundamental group of a graph of groups relative to a maximal tree)

Proof

technique · direct

1.1

Because the action is free, every vertex and edge stabilizer in the quotient graph of groups is trivial. By [L1], it is therefore enough to compute the fundamental group of a graph of trivial groups.

L1given

2.1

With all stabilizers trivial, the path-group relations reduce to $\overset{ˉ}{e} = e^{- 1}$ and there are no vertex-group generators. After killing the maximal-tree edges as in [L3], the remaining generators are exactly the non-tree oriented edges, with no further relations. That is the free-group universal property of [L2].

L2L3step 1.1

3.1

Hence $G$ is free on the non-tree edge generators of the quotient graph.

step 2.1∎

Depends on

Used by

The fundamental group of a graph with trivial groups is free Corollary
A free action and the quotient-graph basis Example
A graph of finite groups giving a virtually free group Example

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. Loh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)
John Meier, Groups, Graphs and Trees (standard reference, not scraped)