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 group (mathematics), groups and topology, topological and geometry, geometric pro ...

, a graph of groups is an object consisting of a collection of

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

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

monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a 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 Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...

graph of groups. It admits an orientation-preserving

action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...

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 r ...

and the

stabilizer subgroup In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

s. This theory, commonly referred to as

Bass–Serre theory Bass–Serre theory is a part of the Mathematics, mathematical subject of group theory that deals with analyzing the algebraic structure of Group (math), groups Group action (mathematics), acting by automorphisms on simplicial Tree (graph theory), t ...

, is due to the work of

Hyman Bass Hyman Bass (; born October 5, 1932). The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the c ... References External links *Directory page at University of MichiganAuthor profilein the database zbMATH {{DEFAUL ...

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 ina ...

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 not ...

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 In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial functi ...

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 i ...

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 an ...

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, an ...

corresponds to an

Generalisations

The simplest possible generalisation of a graph of groups is a 2-dimensional

complex of groups Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

. 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 Cocompact may refer to: * Cocompact group action * Cocompact Coxeter group * Cocompact embedding * Cocompact lattice {{dab ...

properly discontinuous In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...

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 set ...

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 building In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Bu ...

s; their general definition and continued study have been inspired by the ideas of

Gromov Gromov (russian: Громов) is a Russian male surname, its feminine counterpart is Gromova (Громова). Gromov may refer to: * Alexander Georgiyevich Gromov (born 1947), Russian politician and KGB officer * Alexander Gromov (born 1959), R ...

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 Instit ...

from "arbres, amalgames, SL₂", written with the collaboration of