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

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The Bass-Serre coset graph is a tree

Statement

The Bass-Serre coset graph of a graph of groups is a simplicial tree.

Facts & Assumptions

Given: A graph of groups $G$ , a maximal subtree $T$ , and its Bass-Serre coset graph $\tilde{X}$ .

[L1]

The vertices and edges of $\tilde{X}$ are the cosets prescribed in the Bass-Serre construction. (The Bass-Serre tree of a graph of groups)

[L2]

In the graph-of-groups fundamental group, every element has a reduced normal form and a reduced word with positive edge length is nonidentity. (Normal form for the fundamental group of a graph of groups)

Proof

technique · direct

1.1

The graph $\tilde{X}$ is connected. Indeed, a reduced graph-of-groups word for $γ \in π_{1} (G, T)$ records an edge path from the base coset $G_{v_{0}}$ to the vertex $γ G_{v}$ , so every vertex is reached from a base vertex by some path in $\tilde{X}$ .

L1L2given

2.1

A reduced closed path in $\tilde{X}$ based at a vertex coset determines a reduced graph-of-groups word of positive edge length representing the identity element, because following the path returns to the initial coset. That contradicts [L2]. Hence $\tilde{X}$ has no nontrivial reduced closed path.

L1L2step 1.1

3.1

Being connected and having no nontrivial reduced closed path, $\tilde{X}$ is a simplicial tree.

step 1.1step 2.1algebra∎

Depends on

Used by

Bass-Serre structure theorem Theorem
The fundamental group acts without inversions on its Bass-Serre tree Theorem

Dependency tree · two levels

9 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

Jean-Pierre Serre, Trees (standard reference, not scraped)