Full groups of Cuntz–Krieger algebras and Higman–Thompson groups

In this paper, we will study representations of the continuous full group $\Gamma_A$ of a one-sided topological Markov shift $(X_A,\sigma_A)$ for an irreducible matrix $A$ with entries in $\{0,1\}$ as a generalization of Higman–Thompson groups $V_N, 1 < N \in {\mathbb{N}}$. We will show that the group $\Gamma_A$ can be represented as a group $\Gamma_A^{\operatorname{tab}}$ of matrices, called $A$-adic tables, with entries in admissible words of the shift space $X_A$, and a group $\Gamma_A^{\operatorname{PL}}$ of right continuous piecewise linear functions, called $A$-adic PL functions, on $[0,1]$ with finite singularities.