  For an interval exchange map, the number of discontinuities of its iterates
either exhibits linear growth or is bounded. This dichotomy is used to prove
that the group of interval exchanges does not contain distortion elements,
giving examples of groups that do not act faithfully via interval exchanges. As
a further application of this dichotomy, a classification of centralizers in
the group of interval exchanges is given. This classification of centralizers
is used to compute the automorphism group of the interval exchange group.
