Ends of Groups
Let G be a finitely generated group. Let S ⊆ G be a finite generating set of G and let Γ(G, S) be the Cayley graph of G with respect to S. The number of ends of G is defined as e(G) = e(Γ(G, S)). A basic fact in the theory of ends of groups says that e(Γ(G, S)) does not depend on the choice of a finite generating set S of G, so that e(G) is well-defined.
Read more about this topic: Stallings Theorem About Ends Of Groups
Famous quotes containing the words ends of, ends and/or groups:
“In frames as large as rooms that face all ways
And block the ends of streets with giant loaves,
Screen graves with custard, cover slums with praise
Of motor-oil and cuts of salmon, shine
Perpetually these sharply-pictured groves
Of how life should be.”
—Philip Larkin (1922–1986)
“One who is publicly honest about himself ends up by priding himself somewhat on this honesty: for he knows only too well why he is honest—for the same reasons another person prefers illusion and dissimulation.”
—Friedrich Nietzsche (1844–1900)
“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of inclusion and exclusion, conformity and independence.”
—Zick Rubin (20th century)