Groups, Geometry, and Dynamics

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Volume 11, Issue 2, 2017, pp. 499–531
DOI: 10.4171/GGD/405

Published online: 2017-06-26

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

Kengo Matsumoto[1] and Hiroki Matui[2]

(1) Joetsu University of Education, Japan
(2) Chiba University, Japan

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.

Keywords: Higmann–Thompson group, Thompson group, Cuntz–Krieger algebra, topological Markov shift, full group

Matsumoto Kengo, Matui Hiroki: Full groups of Cuntz–Krieger algebras and Higman–Thompson groups. Groups Geom. Dyn. 11 (2017), 499-531. doi: 10.4171/GGD/405