Counting maps from a surface to a graph
Mladen Bestvina, Mark Feighn
Abstract
Let F be a non-abelian finite rank free group, and let Hg be the fundamental group of a surface of genus g with one boundary component represented by Dg in Hg. So, Hg is the free group <a1,b1,...,ag,bg> and Dg is the product of commutators [a1,b1]...[ag,bg]. Given x in F, we are interested in the number num(x) of primitive, i.e. root-free, images of monomorphisms (Hg,Dg) -> (F,x). Our main result is that f(g) >= 2g 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.
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