  In an evolutionary system in which the rules of mutation are local in nature,
the number of possible outcomes after $m$ mutations is an exponential function
of $m$ but with a rate that depends only on the set of rules and not the size
of the original object. We apply this principle to find a uniform upper bound
for the growth rate of certain groups including the mapping class group. We
also find a uniform upper bound for the growth rate of the number of homotopy
classes of triangulations of an oriented surface that can be obtained from a
given triangulation using $m$ diagonal flips.
