  Consider a hyperbolic group G and a quasiconvex subgroup H of infinite index.
We construct a set-theoretic section s of the quotient map (of sets) from G to
G/H such that s(G/H) is a net in G; that is, any element of G is a bounded
distance from s(G/H). This section arises naturally as a set of points
minimizing word-length in each fixed coset gH. The left action of G on G/H
induces an action on s(G/H), which we use to prove that H contains no infinite
subgroups normal in G.
