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TOPOLOGICAL FULL GROUPS OF $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}C^*$-ALGEBRAS ARISING FROM $\beta $-EXPANSIONS

Part of: Structure and classification of infinite or finite groups Topological dynamics Ergodic theory

Published online by Cambridge University Press:  17 July 2014

KENGO MATSUMOTO  and
HIROKI MATUI
KENGO MATSUMOTO*
Affiliation:
Department of Mathematics, Joetsu University of Education, Joetsu 943-8512, Japan email kengo@juen.ac.jp
HIROKI MATUI
Affiliation:
Graduate School of Science, Chiba University, Inage-ku, Chiba 263-8522, Japan
*
✉kengo@juen.ac.jp
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Abstract

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We introduce a family of infinite nonamenable discrete groups as an interpolation of the Higman–Thompson groups by using the topological full groups of the groupoids defined by $\beta $-expansions of real numbers. They are regarded as full groups of certain interpolated Cuntz algebras, and realized as groups of piecewise-linear functions on the unit interval in the real line if the $\beta $-expansion of $1$ is finite or ultimately periodic. We also classify them by a number-theoretical property of $\beta $.

Keywords

Cuntz algebrasC∗-algebrasHigman–Thompson groupsβ-expansionfull groupssymbolic dynamics

MSC classification

Secondary: 20E32: Simple groups 37A45: Relations with number theory and harmonic analysis 37A55: Relations with the theory of $C^*$-algebras 37B10: Symbolic dynamics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 97 , Issue 2 , October 2014 , pp. 257 - 287
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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