Welcome to MathOverflow

Question

1

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 $1 - L(B \phi)$.

flag	asked Feb 27 2012 at 18:33 Mike Shulman 17.2k●20●70

	This number bears no significance in group theory. – Mark Sapir Feb 28 2012 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. – HW Feb 28 2012 at 18:24

Benjamin Enriquez · Answer 1 · 2012-02-27 18:47:25Z UTC

6

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

answered Feb 27 2012 at 18:47

Benjamin Enriquez
61●1

1

Yes. Does that help answer the question of whether it bears any significance in group theory? – Mike Shulman Feb 27 2012 at 20:02

Lefschetz numbers for homomorphisms of free groups

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.