  Let Mod(S) denote the mapping class group of a compact, orientable surface S.
We prove that finitely generated subgroups of Mod(S) which are not virtually
abelian have uniform exponential growth with minimal growth rate bounded below
by a constant depending only, and necessarily, on S. For the proof, we find in
any such subgroup explicit free group generators which are "short" in any word
metric. Besides bounding growth, this allows a bound on the return probability
of simple random walks.
