# Graph of groups

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In [geometric group theory](/source/Geometric_group_theory), a **graph of groups** is an object consisting of a collection of [groups](/source/Group_(mathematics)) indexed by the vertices and edges of a [graph](/source/Graph_(discrete_mathematics)), together with a family of [monomorphisms](/source/Monomorphism) of the edge groups into the vertex groups. There is a unique group, called the **fundamental group**, canonically associated to each finite [connected](/source/Connectivity_(graph_theory)) graph of groups. It admits an orientation-preserving [action](/source/Group_action) on a [tree](/source/Tree_(graph_theory)): the original graph of groups can be recovered from the [quotient graph](/source/Quotient_graph) and the [stabilizer subgroups](/source/Stabilizer_subgroup). This theory, commonly referred to as [Bass–Serre theory](/source/Bass%E2%80%93Serre_theory), is due to the work of [Hyman Bass](/source/Hyman_Bass) and [Jean-Pierre Serre](/source/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](/source/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](/source/Groupoid) of a graph of groups, as shown by [Higgins (1976)](#CITEREFHiggins1976), is that it is defined independently of base point or tree. Also there is [proved](/source/Mathematical_proof) there a nice [normal form](/source/Normal_form_(mathematics)) for the elements of the fundamental groupoid. This includes normal form theorems for a [free product with amalgamation](/source/Free_product_with_amalgamation) and for an [HNN extension](/source/HNN_extension) ([Bass 1993](#CITEREFBass1993)).

## 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](/source/Disjoint_union) of all [coset](/source/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](/source/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](/source/Free_product_with_amalgamation).

- A graph of groups on a single vertex with a [loop](/source/Loop_(graph_theory)) corresponds to an [HNN extension](/source/HNN_extension).

## Generalisations

The simplest possible generalisation of a graph of groups is a 2-dimensional [complex of groups](/source/Orbifold#Complexes_of_groups). These are modeled on [orbifolds](/source/Orbifold) arising from [cocompact](/source/Coxeter%E2%80%93Dynkin_diagram) [properly discontinuous](/source/Properly_discontinuous) actions of discrete groups on 2-dimensional [simplicial complexes](/source/Simplicial_complex) that have the structure of [CAT(0) spaces](/source/CAT(k)_space). The quotient of the simplicial complex has finite stabilizer 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](/source/Glossary_of_graph_theory) occurring in the [links](/source/Link_(geometry)) of vertices have length at least six. Such complexes of groups originally arose in the theory of 2-dimensional [Bruhat–Tits buildings](/source/Bruhat%E2%80%93Tits_building); their general definition and continued study have been inspired by the ideas of [Gromov](/source/Mikhail_Gromov_(mathematician)).

## See also

- [Bass–Serre theory](/source/Bass%E2%80%93Serre_theory)

- [Right-angled Artin group](/source/Right-angled_Artin_group)

## References

- [Bass, Hyman](/source/Hyman_Bass) (1993), "Covering theory for graphs of groups", *Journal of Pure and Applied Algebra*, **89** (1–2): 3–47, [doi](/source/Doi_(identifier)):[10.1016/0022-4049(93)90085-8](https://doi.org/10.1016%2F0022-4049%2893%2990085-8), [MR](/source/MR_(identifier)) [1239551](https://mathscinet.ams.org/mathscinet-getitem?mr=1239551).

- [Bridson, Martin R.](/source/Martin_Bridson); [Haefliger, André](/source/Andr%C3%A9_Haefliger) (1999), *Metric Spaces of Non-Positive Curvature*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Berlin: Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [3-540-64324-9](https://en.wikipedia.org/wiki/Special:BookSources/3-540-64324-9), [MR](/source/MR_(identifier)) [1744486](https://mathscinet.ams.org/mathscinet-getitem?mr=1744486).

- Dicks, Warren; [Dunwoody, M. J.](/source/Martin_Dunwoody) (1989), *Groups Acting on Graphs*, Cambridge Studies in Advanced Mathematics, vol. 17, Cambridge: Cambridge University Press, [ISBN](/source/ISBN_(identifier)) [0-521-23033-0](https://en.wikipedia.org/wiki/Special:BookSources/0-521-23033-0), [MR](/source/MR_(identifier)) [1001965](https://mathscinet.ams.org/mathscinet-getitem?mr=1001965).

- [Haefliger, André](/source/Andr%C3%A9_Haefliger) (1990), "Orbi-espaces [Orbispaces]", *Sur les groupes hyperboliques d'après Mikhael Gromov (Bern, 1988)*, Progress in Mathematics (in French), vol. 83, Boston, MA: Birkhäuser, pp. 203–213, [ISBN](/source/ISBN_(identifier)) [0-8176-3508-4](https://en.wikipedia.org/wiki/Special:BookSources/0-8176-3508-4), [MR](/source/MR_(identifier)) [1086659](https://mathscinet.ams.org/mathscinet-getitem?mr=1086659)

- Higgins, P. J. (1976), "The fundamental groupoid of a graph of groups", *[Journal of the London Mathematical Society](/source/Journal_of_the_London_Mathematical_Society)*, 2nd Series, **13** (1): 145–149, [doi](/source/Doi_(identifier)):[10.1112/jlms/s2-13.1.145](https://doi.org/10.1112%2Fjlms%2Fs2-13.1.145), [MR](/source/MR_(identifier)) [0401927](https://mathscinet.ams.org/mathscinet-getitem?mr=0401927)

- [Scott, Peter](/source/G._Peter_Scott); [Wall, Terry](/source/C._T._C._Wall) (1979), "Topological Methods in Group Theory", *Homological Group Theory*, London Math. Soc. Lecture Note Ser., vol. 36, Cambridge: Cambridge University Press, pp. 137–203, [ISBN](/source/ISBN_(identifier)) [0-521-22729-1](https://en.wikipedia.org/wiki/Special:BookSources/0-521-22729-1), [MR](/source/MR_(identifier)) [0564422](https://mathscinet.ams.org/mathscinet-getitem?mr=0564422).

- [Serre, Jean-Pierre](/source/Jean-Pierre_Serre) (2003), *Trees*, Springer Monographs in Mathematics, Berlin: Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [3-540-44237-5](https://en.wikipedia.org/wiki/Special:BookSources/3-540-44237-5), [MR](/source/MR_(identifier)) [1954121](https://mathscinet.ams.org/mathscinet-getitem?mr=1954121). Translated by [John Stillwell](/source/John_Stillwell) from "arbres, amalgames, SL2", written with the collaboration of [Hyman Bass](/source/Hyman_Bass), 3rd edition, *astérisque* **46** (1983). See Chapter I.5.

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