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Bowen's notion of sofic entropy is a powerful invariant for classifyingprobability-preserving actions of sofic groups. It can be defined in terms ofthe covering numbers of certain metric spaces associated to such an action, the`model spaces'.  The metric geometry of these model spaces can exhibit various interestingfeatures, some of which provide other invariants of the action. This paperexplores an approximate connectedness property of the model spaces, and uses itgive a new proof that certain groups admit factors of Bernoulli shifts whichare not Bernoulli. This was originally proved by Popa. Our proof covers fewerexamples than his, but provides additional information about this phenomenon.

The geometry of model spaces for probability-preserving actions of sofic groups