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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

Published online by Cambridge University Press:  17 July 2014

KENGO MATSUMOTO*
Affiliation:
Department of Mathematics, Joetsu University of Education, Joetsu 943-8512, Japan email [email protected]
HIROKI MATUI
Affiliation:
Graduate School of Science, Chiba University, Inage-ku, Chiba 263-8522, Japan
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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 $.

Type
Research Article
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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