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**17**

votes

**1**answer

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

**2**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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 $x_1,\dots,x_n$. ...