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

Abstract

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 ${\Gamma }_A^{\mathrm{tab}}$ of matrices, called $A$-adic tables, with entries in admissible words of the shift space $X_A$, and a group ${\Gamma }_A^{\mathrm{PL}}$ of right continuous piecewise linear functions, called $A$-adic PL functions, on $[0,1]$ with finite singularities.