---
tag: 11MVWQ5R31
kind: definition
name: Graph of groups
msc: ["20E08", "20E06"]
defines: DC1CQMD3WK
uses: [00C0BR7D7H, 062D9KXJB1, G5S1M9, 0JYTG5K9Y4, 0JGEQXQ4AC, TM7QS7P7X9]
sources:
  - url: https://en.wikipedia.org/wiki/Graph_of_groups
---

## Definition

A *graph* in the sense of Serre consists of a set $V$ of vertices and a set $E$ of oriented edges.
It comes with a map $e \mapsto \bar e$ from $E$ to $E$ with $\bar{\bar e} = e$ and $\bar e \neq e$, and maps $o, t \colon E \to V$ with $o(\bar e) = t(e)$.
A *graph of groups* $\mathbb{G}$ consists of:

1. a connected [[graph|00C0BR7D7H]] $Y$ in this sense;
2. a [[group|G5S1M9]] $G_v$ for each vertex $v$;
3. a group $G_e$ for each edge $e$, with $G_{\bar e} = G_e$;
4. an injective [[homomorphism|0JYTG5K9Y4]] $\alpha_e \colon G_e \to G_{t(e)}$ for each edge $e$.

Let $T$ be a maximal [[tree|062D9KXJB1]] in $Y$.
Form the [[free product|TM7QS7P7X9]] of the groups $G_v$ and of the free group on symbols $t_e$, $e \in E$.
The *fundamental group* $\pi_1(\mathbb{G}, T)$ is its quotient by the relations

- $t_{\bar e} = t_e^{-1}$ for each edge $e$;
- $t_e \, \alpha_e(g) \, t_e^{-1} = \alpha_{\bar e}(g)$ for each edge $e$ and each $g \in G_e$;
- $t_e = 1$ for each edge $e$ of $T$.

## Remarks

Up to isomorphism, $\pi_1(\mathbb{G}, T)$ does not depend on the choice of $T$, and each $G_v$ embeds in it.

Example: $Y$ has two vertices $u, v$ and one pair of edges $e, \bar e$ from $u$ to $v$, with $G_u = A$, $G_v = B$ and $G_e = C$.
Then $T = Y$, all $t_e$ are trivial, and the relations identify $\alpha_{\bar e}(c) \in A$ with $\alpha_e(c) \in B$; so $\pi_1(\mathbb{G}, T)$ is the amalgamated free product $A *_C B$.

Example: $Y$ has one vertex $v$ and a loop $e$, with $G_v = A$ and $G_e = C$.
Then $T = \set{v}$, and $\pi_1(\mathbb{G}, T)$ is generated by $A$ and $t = t_e$ with $t \, \alpha_e(c) \, t^{-1} = \alpha_{\bar e}(c)$; this is the [[HNN extension|YEYERRY37K]] of $A$ along the isomorphism $\alpha_e(C) \to \alpha_{\bar e}(C)$.

Non-example: if some map $\alpha_e$ is not injective, clause 4 fails, and the data is not a graph of groups; the vertex groups need not embed in the resulting quotient.

When all vertex and edge groups are trivial, $\pi_1(\mathbb{G}, T)$ is the free group on the edges of $Y$ outside $T$, one generator for each pair $e, \bar e$.
This is the fundamental group of $Y$ as a topological graph.