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17
votes
1answer
698 views

What are the relations between conjugates and commutators?

The following algebraic structure came up when I was thinking about invariants of coloured knots. The elements are all elements of a noncommutative free group $F$, and the operations are: $a^b= ...
4
votes
2answers
239 views

Primitive subwords in a free group of rank 2

I am not sure yet about what I exactly need to prove, but I guess I can formulate a rough statement similar to the following: Suppose $w\in F_2$ is a primitive word whose length is big enough. Then ...
5
votes
2answers
416 views

Which polynomials are Fricke polynomials ?

Let me recall the definition which seems the most standard of Fricke polynomials. Let $G$ be the free group with two generators $u,v$. It is not very hard to prove that there exists a unique ...
1
vote
1answer
177 views

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 ...
3
votes
0answers
169 views

Generating set for abelianization of “mod $p$ commutator subgroup” of a free group

Let $F_n$ be a free group on $n$ letters, and fix some prime $p \geq 2$. Define $$K_{n,p}=\text{ker}(F_n \rightarrow H_1(F_n;\mathbb{Z}/p))$$ and $$V_{n,p} = H_1(K_{n,p};\mathbb{Q}).$$ For $x \in ...
4
votes
1answer
215 views

Algorithm for image of a free group homomorphism

Let $G$ and $H$ be finitely generated free groups, and let $f:G\to H$ be a homomorphism specified by giving the images of the generators of $G$. Is there an algorithm which takes such an $f$ and a ...
4
votes
1answer
215 views

How to determine free generators of a closed subgroup of a free pro-$p$-group ?

If $F$ is a free discrete group, then any subgroup $H$ of $F$ is free: this is the well-known theorem of Nielsen-Schreier. Moreover, there is a well-known algorithm, the Nielsen-Schreier method that ...
6
votes
2answers
389 views

A metabelian quotient of a free group

I don't know much about free groups (excepted the very basics), and the following question may be trivial, although it isn't to me. Let $F$ be a free group with $n$ generators $x_1,\dots,x_n$. ...