Augmented group systems and shifts of finite type
Let (G, χ, x) be a triple consisting of a finitely presented groupG, epimorphism χ:G →Z, and distinguished elementx ∈G such that χ(x)=1. Given a finite symmetric groupSr, we construct a finite directed graph Γ that describes the set Φr of representations π: Ker χ →Sr as well as the mapping σx:Φr→Φr defined by (σxϱ)(a) = ϱ(x−1ax) for alla ∈ Ker χ. The pair (Φr,σx has the structure of a shift of finite type, a well-known type of compact 0-dimensional dynamical system. We discuss basic properties and applications of therepresentation shift (Φr,σx), including applications to knot theory.
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