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

TL;DR

This paper explores the structure of the full group associated with Cuntz-Krieger algebras, generalizing Higman-Thompson groups by representing these groups through matrices and piecewise linear functions.

Contribution

It introduces new representations of the full group of a topological Markov shift as matrix groups and piecewise linear functions, extending the Higman-Thompson groups framework.

Findings

Representation as A-adic tables of matrices

Representation as A-adic PL functions on [0,1]

Generalization of Higman-Thompson groups

Abstract

In this paper, we will study presentations 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.