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A Free Product Formula for the Sofic Dimension
Published online by Cambridge University Press: 20 November 2018
Abstract
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
$G$.
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- Research Article
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- Copyright © Canadian Mathematical Society 2015
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