  We develop a theory of convex cocompact subgroups of the mapping class group
MCG of a closed, oriented surface S of genus at least 2, in terms of the action
on Teichmuller space. Given a subgroup G of MCG defining an extension L_G:
  1--> pi_1(S) --> L_G --> G -->1 we prove that if L_G is a word hyperbolic
group then G is a convex cocompact subgroup of MCG. When G is free and convex
cocompact, called a "Schottky subgroup" of MCG, the converse is true as well; a
semidirect product of pi_1(S) by a free group G is therefore word hyperbolic if
and only if G is a Schottky subgroup of MCG. The special case when G=Z follows
from Thurston's hyperbolization theorem. Schottky subgroups exist in abundance:
sufficiently high powers of any independent set of pseudo-Anosov mapping
classes freely generate a Schottky subgroup.
