Bowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions
of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated
to such an action, the ‘model spaces’.
The metric geometry of these model spaces can exhibit various interesting features, some of which provide
other invariants of the action. This paper explores an approximate connectedness property of the model
spaces, and uses it give a new proof that certain groups admit factors of Bernoulli shifts which are not
Bernoulli. This was originally proved by Popa. Our proof covers fewer examples than his, but provides additional
information about this phenomenon.
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