  We study fibers of word maps in finite, profinite, and residually finite
groups. Our main result is that, for any word w in the free group on d
generators, there exists $\epsilon > 0$ such that if G is a residually finite
group with infinitely many non-isomorphic non-abelian upper composition
factors, then all fibers of the word map $w\colon G^d \to G$ have Hausdorff
dimension at most $d-\epsilon$.
  We conclude that the profinite completion of a group G as above satisfies no
probabilistic identity. It is therefore randomly free; namely, for any d > 0,
the probability that d randomly chosen elements freely generate a free subgroup
of G is 1. This solves an open problem of Dixon, Pyber, Seress, and Shalev.
  Additional applications and related results are also established. For
example, combining our results with recent results of Bors, we conclude that a
profinite group in which the set of elements of finite odd order has positive
measure has an open prosolvable subgroup. This may be regarded as a
probabilistic version of the Feit-Thompson theorem.
