Generator-offset property: Difference between revisions

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 ===== Some implications of the above =====
-Suppose Q = (α, β, γ) ≠ R = (α, β’, γ’) are the two k-steps in S that project to P. Then T = (α’, β’’, γ’’) projects to I. Here the values in each component differ by at most 1, and α ≠ α’. Either β’’ = β or β’’ = β’. Assume β’’ = β’. Then γ’’ = γ. The cyclic words Λ<sub>1</sub> = the pattern of α and α’, Λ<sub>2</sub> = the pattern of β and β’, and Λ<sub>3</sub> = the pattern of γ and γ’ must form mosses. We now do know that Σ<sub>2</sub> has one more R than Σ<sub>3</sub>, and thus that:
+Suppose Q = (α, β, γ) ≠ R = (α, β’, γ’) are the two k-steps in S that project to P. Then T = (α’, β’’, γ’’) projects to I. Here the values in each component differ by at most 1, and α ≠ α’. Either β’’ = β or β’’ = β’. Assume β’’ = β’. Then γ’’ = γ. The cyclic words Λ<sub>1</sub> = the pattern of α and α’, Λ<sub>2</sub> = the pattern of β and β’, and Λ<sub>3</sub> = the pattern of γ and γ’ must form mosses, and Λ<sub>1</sub> = α...αα’. We now do know that Σ<sub>2</sub> has one more R than Σ<sub>3</sub>, and thus that:
 Λ<sub>2</sub> is λβ μβ’, 2 ≤ μ ≤ ceil(n/2).