  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.
