Counting maps from a surface to a graph

Mladen Bestvina; Mark Feighn

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Counting maps from a surface to a graph

Mladen Bestvina·Mark Feighn

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Abstract

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 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.

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Mladen Bestvina, Mark Feighn. 2005-05-17. Counting maps from a surface to a graph. https://arxiv.org/abs/math/0505363

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