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Let $G = F_X$ be the free group on a finite set $X$, and $\phi\colon G\to G$ a group homomorphism. Consider the number

$$ \sum_{x\in X} (\text{number of occurrences of the generator $x$ in the word $\phi(x)$}) $$

where occurrences of $x^{-1}$ are counted negatively. Does this number bear any significance in group theory? Note that if $\phi$ is the identity map, the formula above reduces to the rank of $G$.

Background: a finitely generated free group $G$ has a classifying space $B G$ which is Spanier-Whitehead dualizable, hence the endomorphism $B \phi\colon B G \to B G$ has a Lefschetz number. The formula above computes $1 - L(B \phi)$.

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This number bears no significance in group theory. – Mark Sapir Feb 28 '12 at 5:32
If I remember correctly, the following paper uses the Lefschetz fixed point theorem to decompose the homology representations of groups of finite automorphisms of surface groups into irreducibles: . The same idea would work for free groups. – HJRW Feb 28 '12 at 18:24

Isn't this the trace of the endomorphism induced by phi on the abelianization of G, which is the free abelian group over X?

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Yes. Does that help answer the question of whether it bears any significance in group theory? – Mike Shulman Feb 27 '12 at 20:02

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