Graph of groups
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 monomorphism
s 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
.
for and define the fundamental group to be the group generated by the vertex groups and elements for each edge subject to the following conditions:
y-1}} if is the edge with the reverse orientation. φy,1(x)}} for all in . 1}} 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 and for an HNN extension
.
, called the universal covering tree, on which acts. It admits the graph as fundamental domain
. The graph of groups given by the stabiliser subgroups on the fundamental domain corresponds to the original graph of groups.
s arising from cocompact properly discontinuous
actions of discrete groups on 2-dimensional simplicial complex
es 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 building
s; their
general definition and continued study have been inspired by the ideas of Gromov.
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 groups act .Another important...
, a graph of groups is an object consisting of a collection of groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
indexed by the vertices and edges of a graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
, together with a family of monomorphism
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
s 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
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
action on a tree
Tree (graph theory)
In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is 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
Bass–Serre theory
Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees...
, is due to the work of Hyman Bass
Hyman Bass
Hyman Bass is an American mathematician, known for work in algebra and in mathematics education. From 1959-1998 he was Professor in the Mathematics Department at Columbia University, where he is now professor emeritus...
and Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
.
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 treeSpanning tree
Spanning tree can refer to:* Spanning tree , a tree which contains every vertex of a more general graph* Spanning tree protocol, a protocol for finding spanning trees in bridged networks...
for and define the fundamental group to be the group generated by the vertex groups and elements for each edge subject to the following conditions:
y-1}} if is the edge with the reverse orientation. φy,1(x)}} for all in . 1}} if is an edge in .
This definition is independent of the choice of .
The benefit in defining the fundamental groupoid
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...
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
Normal form
Normal form may refer to:* Normal form * Normal form * Normal form * Normal form In formal language theory:* Beta normal form* Chomsky normal form* Greibach normal form* Kuroda 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
HNN extension
In mathematics, the HNN extension is a basic construction of combinatorial group theory.Introduced in a 1949 paper Embedding Theorems for Groups by Graham Higman, B. H...
.
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 treeTree
A tree is a perennial woody plant. It is most often defined as a woody plant that has many secondary branches supported clear of the ground on a single main stem or trunk with clear apical dominance. A minimum height specification at maturity is cited by some authors, varying from 3 m to...
, called the universal covering tree, on which acts. It admits the graph as fundamental domain
Fundamental domain
In geometry, the fundamental domain of a symmetry group of an object is a part or pattern, as small or irredundant as possible, which determines the whole object based on the symmetry. More rigorously, given a topological space and a group acting on it, the images of a single point under the group...
. 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 extensionHNN extensionIn mathematics, the HNN extension is a basic construction of combinatorial group theory.Introduced in a 1949 paper Embedding Theorems for Groups by Graham Higman, B. H...
.
Generalisations
The simplest possible generalisation of a graph of groups is a 2-dimensional complex of groups. These are modeled on orbifoldOrbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
s arising from cocompact properly discontinuous
Properly discontinuous
In topology and related branches of mathematics, an action of a group G on a topological space X is called proper if the map from G×X to X×X taking to is proper, and is called properly discontinuous if in addition G is discrete...
actions of discrete groups on 2-dimensional simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
es that have the structure of CAT(0) spaces
CAT(k) space
In mathematics, a CAT space is a specific type of metric space. Intuitively, triangles in a CAT space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature k. In a CAT space, the curvature is bounded from above by k...
. 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
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
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 building
Bruhat–Tits building
In mathematics, a building is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces...
s; their
general definition and continued study have been inspired by the ideas of Gromov.