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
Famous quotes containing the words fundamental and/or group:
“Two fundamental literary qualities: supernaturalism and irony.”
—Charles Baudelaire (1821–1867)
“For me, as a beginning novelist, all other living writers form a control group for whom the world is a placebo.”
—Nicholson Baker (b. 1957)