Let T be a spanning tree for Y and define the fundamental group Γ to be the group generated by the vertex groups Gx and elements y for each edge subject to the following conditions:
- y = y-1 if y is the edge y with the reverse orientation.
- y φy,0(x) y-1 = φy,1(x) for all x in Gy.
- y = 1 if y is an edge in T.
This definition is independent of the choice of T.
The benefit in defining the fundamental groupoid of a graph of groups, as shown by Higgins (1976), 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 (Bass 1993).
Read more about this topic: Graph Of Groups
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