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A Free Product Formula for the Sofic Dimension

  • Robert Graham,
    McGill University
  • Mikael Pichot,
    McGill University
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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 \[ s(G)=\mathfrak{h}(G_1^0)s(G_1)+\mathfrak{h}(G_2^0)s(G_2)-\mathfrak{h}(G_3^0)s(G_3) \] where $\mathfrak{h}$ is the normalized Haar measure on $G$.
Keywords: sofic groups, dynamical systems, orbit equivalence, free entropy sofic groups, dynamical systems, orbit equivalence, free entropy
MSC Classifications: 20E06 show english descriptions Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E06 - Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
 

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