Van Kampen Diagram - Van Kampen Lemma

A key basic result in the theory is the so-called van Kampen lemma which states the following:

  1. Let be a van Kampen diagram over the presentation (†) with boundary label w which is a word (not necessarily freely reduced) in the alphabet A ∪ A−1. Then w=1 in G.
  2. Let w be a freely reduced word in the alphabet A ∪ A−1 such that w=1 in G. Then there exists a reduced van Kampen diagram over the presentation (†) whose boundary label is freely reduced and is equal to w.

Read more about this topic:  Van Kampen Diagram

Famous quotes containing the word van:

    “I passed a tomb among green shades
    Where seven anemones with down-dropped heads
    Wept tears of dew upon the stone beneath.”
    —Unknown. The Thousand and One Nights.

    AWP. Anthology of World Poetry, An. Mark Van Doren, ed. (Rev. and enl. Ed., 1936)