Small Cancellation Theory - Main Definitions - Examples

Examples

  • Let be the standard presentation of the free abelian group of rank two. Then for the symmetrized closure of this presentation the only pieces are words of length 1. This symmetrized form satisfies the C(4)-T(4) small cancellation conditions and the C'(λ) condition for any 1 > λ > 1/4.
  • Let, where k ≥ 2, be the standard presentation of the fundamental group of a closed orientable surface of genus k. Then for the symmetrization of this presentation the only pieces are words of length 1 and this symmetrization satisfies the C'(1/7) and C(8) small cancellation conditions.
  • Let . Then, up to inversion, every piece for the symmetrized version of this presentation, has the form biabj or bi, where 0 ≤ i,j ≤ 100. This symmetrization satisfies the C'(1/20) small cancellation condition.
  • If a symmetrized presentation satisfies the C'(1/m) condition then it also satisfies the C(m) condition.
  • Let rF(X) be a nontrivial cyclically reduced word which is not a proper power in F(X) and let n ≥ 2. Then the symmetrized closure of the presentation satisfies the C(2n) and C'(1/n) small cancellation conditions.

Read more about this topic:  Small Cancellation Theory, Main Definitions

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