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A Free Product Formula for the Sofic Dimension

Published online by Cambridge University Press:  20 November 2018

Robert Graham  and
Mikael Pichot
Robert Graham
Affiliation:
Department of Mathematics and Statistics, McGill University, Montréal, QC H3A 0B9. e-mail: robert.graham2@mail.mcgill.ca, pichot@math.mcgill.ca
Mikael Pichot
Affiliation:
Department of Mathematics and Statistics, McGill University, Montréal, QC H3A 0B9. e-mail: robert.graham2@mail.mcgill.ca, pichot@math.mcgill.ca
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Abstract

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It is proved that if $G\,=\,{{G}_{1}}\,{{*}_{{{G}_{3}}}}\,{{G}_{2}}$ is free product of probability measure preserving $s$–regular ergodic discrete groupoids amalgamated over an amenable subgroupoid ${{G}_{3}}$, then the sofic dimension $s(G)$ satisfies the equality

$$s(G)=\mathfrak{h}(G_{1}^{0})s({{G}_{1}})+\mathfrak{h}(G_{2}^{0})s\left( {{G}_{2}} \right)-\,\mathfrak{h}(G_{3}^{0})s({{G}_{3}}),$$

where $\mathfrak{h}$ is the normalized Haar measure on $G$.

Keywords

20E06.sofic groupsdynamical systemsorbit equivalencefree entropy
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 2 , April 2015 , pp. 369 - 403
Copyright
Copyright © Canadian Mathematical Society 2015

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