# Lefschetz numbers for homomorphisms of free groups

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: arxiv.org/abs/0905.3002 . The same idea would work for free groups. –  HJRW Feb 28 '12 at 18:24