In [[geometric group theory]], a '''graph of groups''' is an object consisting of a collection of [[group (mathematics)|groups]] indexed by the vertices and edges of a [[graph (discrete mathematics)|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 [[connectivity (graph theory)|connected]] graph of groups. It admits an orientation-preserving [[group action|action]] on a [[tree (graph theory)|tree]]: the original graph of groups can be recovered from the [[quotient graph]] and the [[stabilizer subgroup]]s. 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 {{mvar|Y}} is an assignment to each vertex {{mvar|x}} of {{mvar|Y}} of a group {{math|''G''<sub>''x''</sub>}} and to each edge {{mvar|y}} of {{mvar|Y}} of a group {{math|''G''<sub>''y''</sub>}} as well as monomorphisms {{math|φ<sub>''y'',0</sub>}} and {{math|φ<sub>''y'',1</sub>}} mapping {{math|''G''<sub>''y''</sub>}} into the groups assigned to the vertices at its ends.
==Fundamental group== Let {{mvar|T}} be a [[spanning tree]] for {{mvar|Y}} and define the '''fundamental group''' {{math|Γ}} to be the group generated by the vertex groups {{math|''G''<sub>''x''</sub>}} and elements {{mvar|y}} for each edge of {{mvar|Y}} with the following relations:
*{{math|{{overline|''y''}} {{=}} ''y''<sup>−1</sup>}} if {{math|{{overline|''y''}}}} is the edge {{mvar|y}} with the reverse orientation. *{{math|''y'' φ<sub>''y'',0</sub>(''x'') ''y''<sup>−1</sup> {{=}} φ<sub>''y'',1</sub>(''x'')}} for all {{mvar|x}} in {{math|''G''<sub>''y''</sub>}}. *{{math|''y'' {{=}} 1}} if {{mvar|y}} is an edge in {{mvar|T}}.
This definition is independent of the choice of {{mvar|T}}.
The benefit in defining the fundamental [[groupoid]] of a graph of groups, as shown by {{harvtxt|Higgins|1976}}, is that it is defined independently of base point or tree. Also there is [[mathematical proof|proved]] there a nice [[normal form (mathematics)|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]] {{harv|Bass|1993}}.
==Structure theorem== Let {{math|Γ}} be the fundamental group corresponding to the spanning tree {{mvar|T}}. For every vertex {{mvar|x}} and edge {{mvar|y}}, {{math|''G''<sub>''x''</sub>}} and {{math|''G''<sub>''y''</sub>}} can be identified with their images in {{math|Γ}}. It is possible to define a graph with vertices and edges the [[disjoint union]] of all [[coset]] spaces {{math|Γ/''G''<sub>''x''</sub>}} and {{math|Γ/''G''<sub>''y''</sub>}} respectively. This graph is a tree, called the '''universal covering tree''', on which {{math|Γ}} acts. It admits the graph {{mvar|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 (graph theory)|loop]] corresponds to an [[HNN extension]].
==Generalisations== The simplest possible generalisation of a graph of groups is a 2-dimensional [[orbifold#Complexes of groups|complex of groups]]. These are modeled on [[orbifold]]s arising from [[Coxeter–Dynkin diagram|cocompact]] [[properly discontinuous]] actions of discrete groups on {{nowrap|2-dimensional}} [[simplicial complex]]es that have the structure of [[CAT(k) space|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 [[Glossary of graph theory|circuits]] occurring in the [[link (geometry)|links]] of vertices have length at least six. Such complexes of groups originally arose in the theory of {{nowrap|2-dimensional}} [[Bruhat–Tits building]]s; their general definition and continued study have been inspired by the ideas of [[Mikhail Gromov (mathematician)|Gromov]].
== See also == * [[Bass–Serre theory]] * [[Right-angled Artin group]]
== References == *{{citation | last = Bass | first = Hyman | authorlink = Hyman Bass | doi = 10.1016/0022-4049(93)90085-8 | issue = 1-2 | journal = Journal of Pure and Applied Algebra | mr = 1239551 | pages = 3–47 | title = Covering theory for graphs of groups | volume = 89 | year = 1993| doi-access = free }}. *{{citation | last1 = Bridson | first1 = Martin R. | author1-link = Martin Bridson | last2 = Haefliger | first2 = André | author2-link = André Haefliger | isbn = 3-540-64324-9 | location = Berlin | mr = 1744486 | publisher = Springer-Verlag | series = Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | title = Metric Spaces of Non-Positive Curvature | volume = 319 | year = 1999}}. *{{citation | last1 = Dicks | first1 = Warren | last2 = Dunwoody | first2 = M. J. | author2-link = Martin Dunwoody | isbn = 0-521-23033-0 | location = Cambridge | mr = 1001965 | publisher = Cambridge University Press | series = Cambridge Studies in Advanced Mathematics | title = Groups Acting on Graphs | volume = 17 | year = 1989}}. *{{citation | last = Haefliger | first = André | authorlink = André Haefliger | contribution = Orbi-espaces [Orbispaces] | isbn = 0-8176-3508-4 | language = French | location = Boston, MA | mr = 1086659 | pages = 203–213 | publisher = Birkhäuser | series = Progress in Mathematics | title = Sur les groupes hyperboliques d'après Mikhael Gromov (Bern, 1988) | volume = 83 | year = 1990}} *{{citation | last = Higgins | first = P. J. | issue = 1 | journal = [[Journal of the London Mathematical Society]] | mr = 0401927 | pages = 145–149 | series = 2nd Series | title = The fundamental groupoid of a graph of groups | volume = 13 | year = 1976 | doi=10.1112/jlms/s2-13.1.145}} *{{citation | last1 = Scott | first1 = Peter | author1-link = G. Peter Scott | last2 = Wall | first2 = Terry | author2-link = C. T. C. Wall | contribution = Topological Methods in Group Theory | isbn = 0-521-22729-1 | location = Cambridge | mr = 0564422 | pages = 137-203 | publisher = Cambridge University Press | series = London Math. Soc. Lecture Note Ser. | title = Homological Group Theory | volume = 36 | year = 1979}}. *{{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]] from "arbres, amalgames, SL<sub>2</sub>", written with the collaboration of [[Hyman Bass]], 3rd edition, ''astérisque'' '''46''' (1983). See Chapter I.5.
[[Category:Geometric group theory]]