Generator-offset property: Difference between revisions
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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’. 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’. | 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’. 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’. | ||
Suppose Λ<sub>2</sub> is λβ μβ’. Then Λ<sub>3</sub> is a (λ | Suppose Λ<sub>2</sub> is λβ μβ’. Then Λ<sub>3</sub> is a (λ ± 1)γ (μ ∓ 1)γ’ mos. Since neither Λ<sub>2</sub> nor Λ<sub>3</sub> are multimosses, and at least one of μ and (μ ∓ 1) are even, it is now immediate that n is odd. | ||
Either β’’ = β or β’’ = β’. Assume β’’ = β’. Then γ’’ = γ, and Λ<sub>3</sub> is | Either β’’ = β or β’’ = β’. Assume β’’ = β’. Then γ’’ = γ, and Λ<sub>3</sub> is (λ + 1)γ (μ − 1)γ’. Also assume that the first k-step in Σ is Q: | ||
1 … n | 1 … n | ||
Σ = Q W(Q, R) T | Σ = Q W(Q, R) T | ||