Graph of groups

From Wikipedia, the free encyclopedia

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 injective homomorphisms of the edge groups into the vertex groups. There is a single 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 is due to Bass and Serre.

Contents

[edit] Definition

A graph of groups over a graph Y is an assignment of a group Gx to each vertex x and a group Gy to each edge y of Y, as well as injective homomorphisms \varphi_{y,0} and \varphi_{y,1} for each y mapping Gy to the group at each of its endpoints.

[edit] 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 Gx and elements y for each edge subject to the following conditions:

  • \overline{y} = y^{-1} if \overline{y} is the edge y with the reverse orientation.
  • y \varphi_{y,0}(x) y^{-1} = \varphi_{y,1}(x).
  • y = 1 if y is an edge in T.

This definition is independent of the choice of T.

[edit] Structure theorem

Let Γ be the fundamental group corresponding to the spanning tree T. For every vertex x and edge y, Gx and Gy 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/Gx and Γ/Gy 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.

[edit] Examples

[edit] 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.

[edit] See also

[edit] References

  • Serre, Jean-Pierre, Trees, Springer (2003) ISBN 3-540-44237-5 (English translation of "arbres, amalgames, SL2", written with the collaboration of Hyman Bass, 3rd edition, astérisque 46 (1983)). See Chapter I.5.
  • Haefliger, André, Orbi-espaces, (Orbispaces) in Sur les groupes hyperboliques d'après Mikhael Gromov (Bern, 1988), 203--213, Progr. Math., 83, Birkhäuser (1990). ISBN 0-8176-3508-4
  • Bridson, Martin R.; Haefliger, André, Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften, 319. Springer-Verlag, Berlin, 1999. xxii+643 pp. ISBN 3-540-64324-9 MR1744486