# Questions tagged [free-groups]

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**9**

votes

**2**answers

446 views

### A question on the fundamental group of a compact orientable surface of genus >1

Let $G=\pi(X,x)$ be the fundamental group of a compact orientable
surface of genus $g\ge 2$. It is well known that a presentation of
$G$ is
$$G=\langle x_1,y_1,\dots,x_g,y_g \ | \ [x_1,y_1]\cdots
[x_g,...

**11**

votes

**1**answer

386 views

### Is there a name of semidirect product of a group with its automorphism group?

Consider the construction $G \rtimes \text{Aut}(G)$. Here $
G$ is a group, $\text{Aut}(G)$ is the automorphism group and the semidirect product is over the most obvious action.
1) Is there any name ...

**2**

votes

**0**answers

95 views

### Projective G-group

Let $G$ be a fixed group.
Can there be projective $G$-groups which are not free $G$-groups?
If yes, for which groups $G$ it happens?
By a "projective $G$-group", I mean a projective object in the ...

**5**

votes

**1**answer

176 views

### Nielsen-Schreier with operations

The Nielsen-Schreier theorem states that subgroups of a free subgroup are free.
Is this hold also for groups with operations?
Explicitly, let $G$ be a fixed group. Let $F$ be a group with $G$-action ...

**12**

votes

**1**answer

225 views

### Equivalence of surjections from a surface group to a free group

Let $g \geq 2$. Let $S = \langle a_1,b_2,...,a_g,b_g | [a_1,b_1] \cdots [a_g,b_g] \rangle$ be the fundamental group of a genus $g$ surface and let $F_g$ be a free group with $g$ generators. Given ...

**6**

votes

**4**answers

525 views

### What is a geodesic in Outer space?

The Culler-Vogtmann Outer space $\text{CV}_n$ is an analogue of Teichmuller space for the group $\text{Out}(F_n)$.
Is there any notion of a geodesic path in $\text{CV}_n$? Are there different ...

**4**

votes

**1**answer

174 views

### Database subgroups of free group

Is there some database that contains "all" low-index normal subgroups of the free group on two generators?
Extension: does there exist such a GAP-database?
Thank you!

**6**

votes

**0**answers

97 views

### Localizations of group algebras of free groups

$\newcommand{\QQ}{\Bbb Q}$
Let $G$ be a free group on the symbols $x_1, \dots, x_n$, with $\QQ[G]$ its rational group algebra.
Write $\varepsilon: \QQ[G] \to \QQ$ for the augmentation, and for $\...

**5**

votes

**1**answer

109 views

### Dense abstract free subgroups in a free profinite group

Let $\langle a, b \rangle = F_2$ be a two-generator free group and $\hat{F_2}$ be its profinite completion. Is there an element $c\in \hat{F_2}$ such that $\langle a, b, c\rangle \le \hat{F_2}$ is ...

**5**

votes

**1**answer

235 views

### $\operatorname{Out}(F_n)$ is not linear for $n > 3$

The paper The Tits alternative for $\operatorname{Out}(F_n)$ I by Bestvina, Feighn and Handel and the paper Automorphisms of free groups and Outer space by Vogtmann both state that $\operatorname{Aut}(...

**9**

votes

**1**answer

366 views

### The Tits alternative for $\operatorname{Out}(F_n)$

Not sure if this is the right place to ask this, but the paper I am reading seems to be too specialised for mathstack (if you do not agree, pleas let me know and I will take down this question)
I am ...

**8**

votes

**2**answers

236 views

### Equations in free groups satisfying all elements

please help me to solve the following problem.
Let $F$ be a non-abelian free group and $w(x)=1$ be an equation in one variable $x$ ($w(x)$ may contain elements of $F$ as constants). Clearly, one can ...

**6**

votes

**1**answer

314 views

### An algorithm determining whether two subgroups of a finitely generated free group are automorphic

In the book Lyndon, Schupp, Combinatorial Group Theory, P.30 in the edition from 2000 They mention an unpublished work by Waldhausen that is said to give an algorithm to determine whether two ...

**5**

votes

**1**answer

183 views

### Relation between commutator length and stable commutator length in free groups

In Bardakov, Algebra and Logic, Vol. 39, No. 4, 2000 I have found the following (page 225, see https://link.springer.com/article/10.1007/BF02681648)
We pronounce tile validity of the following:
...

**8**

votes

**2**answers

334 views

### An endomorphism of free groups

So you have a free group $F_n$, freely generated by $\alpha_1 \cdots \alpha_n$. Pick any $n$ elements $g_1 \cdots g_n$ and define an endomorphism $\psi$ of $F_n$ by $\psi(\alpha_i) = g_i^{-1}\...

**2**

votes

**0**answers

176 views

### Is a matrix group free?

Let two matrices $P = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$ and $S = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 1 & 1 \...

**11**

votes

**1**answer

250 views

### “Bisecting” a free subgroup with respect to word length

My broad question is regarding the lengths of (reduced) words in a subgroup of a free group.
As motivation, consider the free group $Gp(S)$ where $|S|=n$, that is, a free group of rank $n$. Let $S=\{...

**7**

votes

**3**answers

301 views

### growth of a free group automorphism is same for finite index subgroups?

Let $X=\{x_1,\dots,x_N\}$ and $F=F(X)$ be a free group generated by $X$. Let $\phi\colon F\to F$ be an automorphism of $F$. Define a growth function of $\phi$ as:
$$
\operatorname{gr}_{\phi,X}(n)=\...

**3**

votes

**1**answer

101 views

### Maximal power in a sequence of iterated commutators in the rank two free group

I have the following problem: in the free group $F_2=\langle a,b\rangle$, we define the sequence
$\begin{cases}
w_0=a, \\
w_1=b, \\
w_{n+2}=[w_{n+1},w_{n}] & \text{for }n\ge 0.
\end{cases}$
So $...

**9**

votes

**1**answer

140 views

### Detecting/Characterising positive elements in free groups

Let $X$ be a set, and let $F(X)$ be the free group generated by $X$.
I will say that an element of $F(X)$ is positive if it is in the monoid generated by all the conjugates in $F(X)$ of every member ...

**2**

votes

**0**answers

118 views

### Concentration of Reduced words

This might be a rather broad question, and I'll be satisfied with some intuition and pointers to relevant literature. However, I'll certainly fill in more context and details based on any feedback.
...

**4**

votes

**1**answer

226 views

### Finite index subgroups of a RAAG

Let $G$ be the group given by the presentation
$$\langle x,y,z,w \ | \ xy = yx, yz = zy, zw = wz\rangle.$$
This is a right-angled Artin group (RAAG) whose graph is a path on $4$ vertices.
We can ...

**2**

votes

**2**answers

263 views

### Subgroup of a free group that is characteristic but not totally characteristic

Looking for a counter example (if it exists) and a reference for further reading. Can there be a subgroup of finite index in a finitely generated free group that is characteristic but not totally ...

**7**

votes

**0**answers

140 views

### Structure constants of Lyndon-Shirshov basis of the free Lie ring

Let $X$ be an alphabet, ${\sf Lyn}$ be the set of Lyndon words on $X$ and $L$ be the free Lie ring on $X.$ For $w\in {\sf Lyn}$ we denote by $[w]$ the corresponding element of the Lyndon-Shirshov ...

**2**

votes

**0**answers

78 views

### Automorphisms of a free topological product

Let $G$, $G_1$, $G_2$ be Hausdorff topological groups. I am mainly interested in the case when those groups are profinite.
Let $G$ act continuously on $G_1$ and $G_2$ via continuous automorphisms, i.e....

**4**

votes

**0**answers

91 views

### Do the “Nielsen” IA-automorphisms of a profinite free group $\widehat{F}$ of rank 2 form a normal subgroup of $\mathrm{Aut}(\widehat{F})$?

Let $F$ be the discrete free group of rank 2, and let $\widehat{F}$ be its profinite completion, equipped with an embedding
$$i : F\hookrightarrow\widehat{F}$$
By a result of Asada, this embedding ...

**9**

votes

**1**answer

436 views

### Normal closures of finitely generated subgroups of a free group

Is it true that for every finitelty generated subgroup $H$ of infinite index in a free
group $F$ on the two letters $\{x,y\}$, there exists a finite index
subgroup $K$ of $H$, such that the normal ...

**13**

votes

**2**answers

299 views

### Free groups and free restricted Lie algebras

If $G$ is any group and $\gamma_k(G)$ denotes the $k$th term in the lower central series of $G$, then the commutator bracket on $G$ endows
$$\mathcal{L}(G) = \bigoplus_{k=1}^{\infty} \gamma_k(G) / \...

**7**

votes

**2**answers

612 views

### Is there a useful limit or co-limit of a diagram that has only a single object?

I'm starting to study category theory kind of informally and everytime I read about the definitions of limits and co-limits, the first three examples are always the same:
terminal/initial objects,
...

**1**

vote

**1**answer

79 views

### Under what condition can any $X\in GL_2(R)$ be reduced to a triangular matrix?

Suppose $R$ is a (possibly noncommutative) ring. I was thinking of $R=S[x_1,\ldots,x_n]$ or $R=S[x_1,x_1^{-1},\ldots,x_n,x_n^{-1}]$ for $S$ some (possibly noncommutative) ring. Now, let $GL_2(R)$ be ...

**3**

votes

**2**answers

203 views

### Free subgroup of a quotient

Let $F$ be a free group on $x,y,z$. Fix $n>1$ (I am ready to assume that $n$ is large enough). Let $\mathcal{W}$ be the set of cyclically reduced words $w$ in $F$ where the letter $z$ appears at ...

**14**

votes

**2**answers

719 views

### A result of Schützenberger on commutators and powers in free groups

It is an old result of Schützenberger that in a free group, a basic commutator cannot be a proper power. A look at the original reference
M.-P. Schützenberger, Sur l'équation $a^{2+n} = b^{2+m}c^{2+p}...

**1**

vote

**2**answers

164 views

### Examples of IF-groups

I have seen that several authors say that an infinite group $G$ is an IF-group (or has the IF-property) if every subgroup of infinite index in $G$ is free (for instance, see https://arxiv.org/pdf/1607....

**0**

votes

**0**answers

100 views

### specific qi on free groups

Let $F_n$ be the free group on $n$ generators, $n>1$.
If $\phi$ is a quasi-isometry (or a bijective bilipschitz equivalence) on $F_n$, then what can we say about the explicit form of $\phi$?
In ...

**1**

vote

**1**answer

90 views

### Free algebras on sets of different cardinality - for what theories are they non-isomorphic?

Following the case of groups, I asked in this MSE question for a quick proof that given a free-forgetful adjunction $F\dashv U$ for some algebraic theory, we have $X\not\cong Y\implies FX\not\cong FY$....

**5**

votes

**0**answers

138 views

### free subgroups of $SL_2(\mathbb{Z[i]})$

The group $SL_2(\mathbb{Z})$ contains many free subgroups, for example all of the principal congruence subgroups for $n\geq 3$ and the subgroup $\left\langle \left(\begin{array}{cc}
1 & 2\\
0 &...

**4**

votes

**2**answers

255 views

### Is the mapping torus of an automorphism of a free group virtually an amalgamated product?

Let $F$ be a nonabelian finitely generated free group,
let $\tau \in \mathrm{Aut}(F)$ be an element of infinite order,
and set $G = F \rtimes \mathbb{Z}$,
where the action of $\mathbb{Z}$ on $F$ is ...

**10**

votes

**1**answer

341 views

### The set of subgroups of $F_2$

This question came up in our algebraic topology class and our Professor didn't know the answer. I also couldn't find an answer so far.
What is the cardinality of the set of subgroups of $F_2$?
...

**6**

votes

**2**answers

210 views

### Finding an “optimal” quotient in a free group

Consider the abelian free group $G = \mathbf{Z}^n$ of rank $n$ and a finite subset $A \subset G \setminus \{0\}$. Since $G$ is residually finite, there is a subgroup $H \subset G$ such that $A \cap H =...

**4**

votes

**1**answer

252 views

### finitely presented subgroup and free solvable group of class 3

Let $F(n)$ be free group of rank $n\geq 2$. Denote by $F_d(n)$ the d-th derived subgroup, that is $F_d(n)=[F_{d-1}(n),F_{d-1}(n)]$ where $F_0(n)=F(n)$. The free solvable group of rank $n$ and ...

**4**

votes

**1**answer

118 views

### Genericity of irreducible automorphisms of free groups

I have seen in the literature that Irreducible outer automorphisms of a free group $F_n$ are "generic".
I would like to ask that if for example : it is true that for any generating set $X$ of $Out(...

**3**

votes

**1**answer

170 views

### Wild automorphisms of profinite groups

Is there a profinite group $G$, a continuous automorphism $\alpha$ of
$G$ and a topologically finitely generated closed subgroup $H \leq G$
such that $\alpha(H) \lneq H$ ?
Note that if an example ...

**2**

votes

**0**answers

77 views

### Fully residually free groups and completion

Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?

**6**

votes

**0**answers

218 views

### Wild automorphisms of a free group

Let $F_X$ be a free group on a countably infinite set $X$. Let $\alpha$ be an automorphism of $F_X$ and $H$ a closed subgroup of $F_X$ in the profinite topology.
Is it possible that $\alpha(H) \...

**2**

votes

**0**answers

120 views

### Rank gradient in free products amalgamating a finite subgroup

Let $A,B$ be finitely generated groups with a common finite subgroup $C$. Suppose that $[A : C] > 2, [B : C] > 1$.
Must $A *_C B$ have positive rank gradient?
See Which 3-manifolds have ...

**2**

votes

**0**answers

104 views

### Salvaging Howson's theorem for free profinite groups

This is an attempt to find a correct version of Do free profinite groups satisfy Howson's theorem?
Let $F$ be a free profinite group, and let $A,B \leq F$ be closed finitely generated subgroups ...

**9**

votes

**3**answers

356 views

### Integer matrix that does not belong to a free group of rank 2

I'm given two matrices in $SL_2(\mathbb{Z})$
$$
A = \left(\begin{array}{cc}
2 & 3\\
3 & 5
\end{array}\right), \ \
B = \left(\begin{array}{cc}
5 & 3\\
3 &...

**4**

votes

**0**answers

129 views

### Example of “exotic” verbal subgroups of free groups

This will be an ambiguous question.
I am interested in various examples that appear in the literature of verbal subgroups of free groups, but which are not part of the "classical examples" like ...

**4**

votes

**2**answers

223 views

### On the Magnus Representation of Free Metabelian Group

Let $F=\langle x_1,x_2\rangle$ be a free group of rank $2$ and $\Phi=F/F''=\langle \overline{x}_1, \overline{x}_2\rangle$ where $F''$ is second derived subgroup of $F$ (i.e. $F'=[F,F]$ and $F''=[F',F']...

**9**

votes

**1**answer

270 views

### Automorphism groups for free groups with action

Let $A$ be a (finitely generated) free group and $G$ be a (finitely generated) free $A$-group - that is, a group with an action of $A$, which is free in the category of groups with an $A$-action. ...