A famous question of Gromov asks whether every hyperbolic
group contains a subgroup which is isomorphic to the fundamental group of
a closed surface. Surface subgroups play a very important role in many
areas of low-dimensional topology, for example in Agol's recent proof that
every hyperbolic 3-manifold has a finite cover which fibers over the circle. I
would like to describe several ways to build surface subgroups in certain
hyperbolic groups. The role of hyperbolicity is twofold here: first,
hyperbolic geometry allows one to certify injectivity by *local* data;
second, hyperbolic dynamics allows one to use ergodic theory to
produce the pieces out of which an injective surface can be built. I would
like to sketch a proof of the fact that the extension of a free group
associated to a ''random'' endomorphism contains a surface subgroup with
probability one. This is joint work with Alden Walker.