Show that a finite graph product of finitely generated groups is finitely generated, and similarly for finitely presented groups.
Solution
Let \(\Gamma\) be a finite connected graph of groups and choose a maximal tree \(T\) in its underlying graph.
1. If all vertex groups are finitely generated, then the graph product \(\pi_1(K\Gamma)\) is finitely generated.
For each vertex \(v\), choose a finite generating set \(S_v\) for the vertex group \(G_v\). The standard graph-of-groups presentation has generators \[ \bigcup_v S_v \] together with one stable letter \(t_e\) for each unoriented edge \(e\) not in the maximal tree \(T\). There are finitely many vertices and finitely many such edges because \(\Gamma\) is finite. Thus this is a finite generating set for \(\pi_1(K\Gamma)\).
2. Suppose now that every vertex group is finitely presented and every edge group is finitely generated.Then the graph product is finitely presented.
1.Impose:
- the finitely many vertex relators \(R_v\);
- for each tree edge \(e\) and each \(c\in C_e\), the relation identifying the two images of \(c\) in the endpoint vertex groups;
- for each non-tree edge \(e\) and each \(c\in C_e\), the HNN relation \[ t_e\,\iota_{e,0}(c)\,t_e^{-1}=\iota_{e,1}(c). \]
These relations suffice for every element of an edge group because they hold on a generating set and both endpoint maps are homomorphisms. Since the graph is finite, only finitely many relations occur. Thus the graph product is finitely presented.
3. In particular, a finite graph product of finitely presented groups is finitely presented.
A finitely presented edge group is finitely generated, so the hypotheses of 2 are satisfied when all groups in the finite graph of groups are finitely presented.