Geodesic
A Free Product Formula for the Sofic Dimension
Canadian journal of mathematics, Tome 67 (2015) no. 2, pp. 369-403

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DOI

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 $G$ .
DOI : 10.4153/CJM-2014-019-5
Mots-clés : 20E06., sofic groups, dynamical systems, orbit equivalence, free entropy 
Graham, Robert; Pichot, Mikael. A Free Product Formula for the Sofic Dimension. Canadian journal of mathematics, Tome 67 (2015) no. 2, pp. 369-403. doi: 10.4153/CJM-2014-019-5
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     journal = {Canadian journal of mathematics},
     pages = {369--403},
     year = {2015},
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     number = {2},
     doi = {10.4153/CJM-2014-019-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2014-019-5/}
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