Graph of groups

In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups. There is a unique group, called the fundamental group, canonically associated to each finite connected graph of groups. It admits an orientation-preserving action on a tree: the original graph of groups can be recovered from the quotient graph and the stabilizer subgroups. This theory, commonly referred to as Bass–Serre theory, is due to the work of Hyman Bass and Jean-Pierre Serre.

A graph of groups over a graph $Y$ is an assignment to each vertex $x$ of $Y$ of a group $G x$ and to each edge $y$ of $Y$ of a group $G y$ as well as monomorphisms $φ y,0$ and $φ y,1$ mapping $G y$ into the groups assigned to the vertices at its ends.

Fundamental group

Let $T$ be a spanning tree for $Y$ and define the fundamental group $Γ$ to be the group generated by the vertex groups $G x$ and elements $y$ for each edge of $Y$ with the following relations:

$y = y -1$ if $y$ is the edge $y$ with the reverse orientation.
$y φ y,0 (x) y -1 = φ y,1 (x)$ for all $x$ in $G y$ .
$y = 1$ if $y$ is an edge in $T$ .

This definition is independent of the choice of $T$ .

The benefit in defining the fundamental groupoid of a graph of groups, as shown by Higgins (1976), is that it is defined independently of base point or tree. Also there is proved there a nice normal form for the elements of the fundamental groupoid. This includes normal form theorems for a free product with amalgamation and for an HNN extension (Bass 1993).

Structure theorem

Let $Γ$ be the fundamental group corresponding to the spanning tree $T$ . For every vertex $x$ and edge $y$ , $G x$ and $G y$ can be identified with their images in $Γ$ . It is possible to define a graph with vertices and edges the disjoint union of all coset spaces $Γ/ G x$ and $Γ/ G y$ respectively. This graph is a tree, called the universal covering tree, on which $Γ$ acts. It admits the graph $Y$ as fundamental domain. The graph of groups given by the stabilizer subgroups on the fundamental domain corresponds to the original graph of groups.

Examples

A graph of groups on a graph with one edge and two vertices corresponds to a free product with amalgamation.
A graph of groups on a single vertex with a loop corresponds to an HNN extension.

Fundamental group

Structure theorem

Examples

Generalisations

See also

References

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