  Let F be a non-abelian finite rank free group, and let H_g be the fundamental
group of a surface of genus g with one boundary component represented by D_g in
H_g. So, H_g is the free group <a_1,b_1,...,a_g,b_g> and D_g is the product of
commutators [a_1,b_1]...[a_g,b_g]. Given x in F, we are interested in the
number num(x) of primitive, i.e. root-free, images of monomorphisms (H_g,D_g)
-> (F,x). Our main result is that f(g) >= 2^g where f(g)=sup num(x), where sup
is taken over all elements x in F. This answers a question of Zlil Sela that is
related to his work on the Tarski problem. We also show that f is independent
of F and go on to obtain similar results where the monomorphisms considered are
additionally required to have minimal genus.
