Graph of groups

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 .

definition

A graph of groups is given by the following data:

• a directed connected graph , so that, for each edge of the upturned edge to part ${\displaystyle (E,K)}$ ${\displaystyle k=(e_{0},e_{1})\in K}$ ${\displaystyle {\overline {k}}=(e_{1},e_{0})}$ ${\displaystyle K}$
• a "corner group" for each corner ${\displaystyle G_{e}}$ ${\displaystyle e\in E}$
• an "edge group" for each edge , so that for all ${\displaystyle G_{k}}$ ${\displaystyle k\in K}$ ${\displaystyle G_{k}=G_{\overline {k}}}$ ${\displaystyle k}$
• injective homomorphisms for each edge ${\displaystyle \alpha _{k,i}\colon G_{k}\to G_{e_{i}},i=0,1}$ ${\displaystyle k=(e_{0},e_{1})\in K}$

Fundamental group

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. ${\displaystyle T}$ ${\displaystyle (E,K)}$

The fundamental group of the graph of groups is defined as the free product

${\displaystyle (\ast _{e\in E}G_{e})\ast F(K)}$

(where the free group is denoted by basis ) modulo of the following relations: ${\displaystyle F(K)}$ ${\displaystyle K}$

• ${\displaystyle {\overline {k}}\alpha _{k,0}(g)k=\alpha _{k,1}(g)}$ for all ${\displaystyle k\in K,g\in G_{k}}$
• ${\displaystyle {\overline {k}}k=1}$ for all ${\displaystyle k\in K}$
• ${\displaystyle k=1}$for all edges occurring in the spanning tree ${\displaystyle T}$

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 ${\displaystyle (E,K)}$ ${\displaystyle k=(e_{0},e_{1})}$ ${\displaystyle e_{0},e_{1}}$

${\displaystyle G_{e_{0}}*_{G_{k}}G_{e_{1}}}$.

• 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 ${\displaystyle (E,K)}$ ${\displaystyle k=(e,e)}$ ${\displaystyle e_{0}=e_{1}}$

${\displaystyle G_{e}*_{\alpha }}$

for the through

${\displaystyle \alpha :=\alpha _{k,1}(\alpha _{k,0})^{-1}\colon \alpha _{k,0}(G_{k})\to \alpha _{k,1}(G_{k})}$

given homomorphism between the subgroups and of . ${\displaystyle \alpha }$ ${\displaystyle \alpha _{k,0}(G_{k})}$ ${\displaystyle \alpha _{k,1}(G_{k})}$ ${\displaystyle G_{e}}$

See also

• Bass Serre Tree

literature

• 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