In mathematics , graphs of groups are a construction of group theory that can be used to construct iterated amalgamated products and HNN extensions and which is important in Bass-Serre theory .
To define the fundamental group of a graph of groups, a spanning tree in the graph must first be selected. The fundamental group is ultimately independent of the chosen spanning tree.
The fundamental group of the graph of groups is defined as the free product
(where the free group is denoted by basis ) modulo of the following relations:
for all
for all
for all edges occurring in the spanning tree
Examples
Let it be the graph consisting of an edge with two vertices . Then the fundamental group of a graph of groups is the amalgamated product
.
Let it be the graph consisting of an edge with two matching corner points (a "loop"). Then the fundamental group of a graph of groups is the HNN extension
Jean-Pierre Serre : Arbres, amalgames, SL 2 . Rédigé avec la collaboration de Hyman Bass. Astérisque, No. 46. Société Mathématique de France, Paris, 1977.
English translation: Trees. Translated from the French original by John Stillwell. Corrected 2nd printing of the 1980 English translation. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. ISBN 3-540-44237-5