# Augmented group systems and shifts of finite type

## Authors

- Received:

DOI: 10.1007/BF02761041

- Cite this article as:
- Silver, D.S. & Williams, S.G. Israel J. Math. (1996) 95: 231. doi:10.1007/BF02761041

## Abstract

Let (*G, χ, x*) be a triple consisting of a finitely presented group*G*, epimorphism χ:*G* →**Z**, and distinguished element*x* ∈*G* such that χ(*x*)=1. Given a finite symmetric group*S*
_{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 all*a* ∈ 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 the*representation shift* (Φ_{
r
},σ_{
x
}), including applications to knot theory.