A Free Product Formula for the Sofic Dimension

Graham, Robert ; Pichot, Mikael

Canadian journal of mathematics, Tome 67 (2015) no. 2, pp. 369-403

Résumé

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 $G$ .

Détail

DOI : 10.4153/CJM-2014-019-5

Mots-clés : 20E06., sofic groups, dynamical systems, orbit equivalence, free entropy

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