Graph of groups

Last updated January 10, 2023

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.

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 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.

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 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 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.

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References

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Bridson, Martin R.; Haefliger, André (1999), Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 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, vol. 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), vol. 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
Scott, Peter; Wall, Terry (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 0-521-22729-1, MR 0564422 .
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, astérisque46 (1983). See Chapter I.5.

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