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 stabiliser subgroups. This theory, commonly referred to as Bass–Serre theory, is due to the work of Hyman Bass and Jean-Pierre Serre.

Definition

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 subject to the following conditions:

• 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 stabiliser 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.

Generalisations

The simplest possible generalisation of a graph of groups is a 2-dimensional complex of groups. These are modeled on orbifolds arising from cocompact properly discontinuous actions of discrete groups on 2-dimensional simplicial complexes that have the structure of CAT(0) spaces. The quotient of the simplicial complex has finite stabiliser groups attached to vertices, edges and triangles together with monomorphisms for every inclusion of simplices. A complex of groups is said to be developable if it arises as the quotient of a CAT(0) simplicial complex. Developability is a non-positive curvature condition on the complex of groups: it can be verified locally by checking that all circuits occurring in the links of vertices have length at least six. Such complexes of groups originally arose in the theory of 2-dimensional Bruhat–Tits buildings; their general definition and continued study have been inspired by the ideas of Gromov.

See also

• Right-angled Artin group

References

• Bass, Hyman (1993), "Covering theory for graphs of groups", Journal of Pure and Applied Algebra 89 (1-2): 3–47, doi:10.1016/0022-4049(93)90085-8, MR 1239551.
• Bridson, Martin R.; Haefliger, André (1999), Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 319, Berlin: Springer-Verlag, ISBN 3-540-64324-9, MR 1744486.
• Dicks, Warren; Dunwoody, M. J. (1989), Groups Acting on Graphs, Cambridge Studies in Advanced Mathematics 17, Cambridge: Cambridge University Press, ISBN 0-521-23033-0, MR 1001965.
• Haefliger, André (1990), "Orbi-espaces [Orbispaces]", Sur les groupes hyperboliques d'après Mikhael Gromov (Bern, 1988), Progress in Mathematics (in French) 83, Boston, MA: Birkhäuser, pp. 203–213, ISBN 0-8176-3508-4, MR 1086659
• Higgins, P. J. (1976), "The fundamental groupoid of a graph of groups", Journal of the London Mathematical Society, 2nd Series 13 (1): 145–149, doi:10.1112/jlms/s2-13.1.145, MR 0401927.
• Serre, Jean-Pierre (2003), Trees, Springer Monographs in Mathematics, Berlin: Springer-Verlag, ISBN 3-540-44237-5, MR 1954121. Translated by John Stillwell from "arbres, amalgames, SL₂", written with the collaboration of Hyman Bass, 3rd edition, astérisque 46 (1983). See Chapter I.5.