Abstract
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 groupS r, we construct a finite directed graph Γ that describes the set Φ r of representations π: Ker χ →S r as well as the mapping σ x :Φ r →Φ r defined by (σ x ϱ)(a) = ϱ(x −1 ax) for alla ∈ Ker χ. The pair (Φ r ,σ x 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 (Φ r ,σ x ), including applications to knot theory.
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Silver, D.S., Williams, S.G. Augmented group systems and shifts of finite type. Israel J. Math. 95, 231–251 (1996). https://doi.org/10.1007/BF02761041
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DOI: https://doi.org/10.1007/BF02761041