geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these group ...

, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

, 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 In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...

: the original graph of groups can be recovered from the

quotient graph In graph theory, a quotient graph ''Q'' of a graph ''G'' is a graph whose vertices are blocks of a partition of the vertices of ''G'' and where block ''B'' is adjacent to block ''C'' if some vertex in ''B'' is adjacent to some vertex in ''C'' with ...

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 Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ...

Definition

A graph of groups over a graph is an assignment to each vertex of of a group and to each edge of of a group as well as monomorphisms and mapping into the groups assigned to the vertices at its ends.

Fundamental group

Let be a

spanning tree In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is no ...

for and define the fundamental group to be the group generated by the vertex groups and elements for each edge of with the following relations: * if is the edge with the reverse orientation. * for all in . * if is an edge in . This definition is independent of the choice of . The benefit in defining the fundamental groupoid of a graph of groups, as shown by , 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 In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and ...

and for an

HNN extension In mathematics, the HNN extension is an important construction of combinatorial group theory. Introduced in a 1949 paper ''Embedding Theorems for Groups'' by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group ''G'' into ano ...

Structure theorem

Let be the fundamental group corresponding to the spanning tree . For every vertex and edge , and can be identified with their images in . It is possible to define a graph with vertices and edges the disjoint union of all coset spaces and respectively. This graph is a

, called the universal covering tree, on which acts. It admits the graph as

fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...

. 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

. *A graph of groups on a single vertex with a

loop Loop or LOOP may refer to: Brands and enterprises * Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live * Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets * Loop Mobile, ...

corresponds to an

Generalisations

The simplest possible generalisation of a graph of groups is a 2-dimensional complex of groups. These are modeled on

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...

s arising from cocompact properly discontinuous actions of discrete groups on

simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...

es 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 Bruhat–Tits buildings; their general definition and continued study have been inspired by the ideas of Gromov.

References

*. *. *. * * *. *{{citation , last = Serre , first = Jean-Pierre , authorlink = Jean-Pierre Serre , isbn = 3-540-44237-5 , location = Berlin , mr = 1954121 , publisher = Springer-Verlag , series = Springer Monographs in Mathematics , title = Trees , year = 2003. Translated by

John Stillwell John Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University. Biography He was born in Melbourne, Australia and lived there until he went to the Massachusetts Ins ...