A Free Product Formula for the Sofic Dimension. (20th April 2015)

• Record Type: Journal Article
• Authors: Graham, Robert; Pichot, Mikael
• Is Part Of: Canadian journal of mathematics. Volume 67:Number 2(2015)
• Page Start: 369
• Page End: 403
• Publication Date: 2015-04-20
• Subjects: 20E06.; sofic groups -- dynamical systems -- orbit equivalence -- free entropy; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2014-019-5
• Languages: English
• ISSNs: 0008-414X

Abstract

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 $G$ .