# Questions tagged [free-groups]

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165
questions

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### Some questions on a paper of Baumslag and Solitar

I was recently trying taking a look at the paper "Some two-generator one-relator non-hopfian groups" by Baumslag and Solitar where they introduce the groups now known as the Baumslag-Solitar ...

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52 views

### Computability of the “free envelope rank” of an endomorphism of a free group

Let $F$ be a free group freely generated by the finite set $S$ and $\sigma\colon F\to F$ be a group morphism. We define the free envelope rank of $\sigma$, written $r(\sigma)$, as the smallest $k$ for ...

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**3**answers

355 views

### Membership to double cosets in free groups

Is there an elementary and efficient algorithm for testing the membership to a double coset of f.g. subgroups in a free group?
Has this membership problem been implemented in GAP/Magma?
More ...

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votes

**1**answer

118 views

### Dirichlet region of a free group

Let $G$ be a non-uniform lattice Fuchsian group and let $P$ be a Dirichlet region for $G$. In particular $G$ has parabolic elements, $P$ is not compact and has finite area. We are in the unit disc. Is ...

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votes

**1**answer

128 views

### Howson property of automorphism group of $F_2$ and of $F_3$

Is the intersection of any two finitely generated subgroups of $\operatorname{Aut}(F_2)$ (resp. $\operatorname{Aut}(F_3)$) again finitely generated? That is, does $\operatorname{Aut}(F_2)$ (resp. $\...

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34 views

### When does bottom stratum of relative train track map give rise to irreducible outer automorphism of free groups

Let $F_n$ be the free group of finite rank $n$ and let $\mathcal{O} \in \text{Out}(F_n)$. Let $\Gamma$ be a finite graph and $f : \Gamma \to \Gamma$ a relative train track representative of $\mathcal{...

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91 views

### How can I build free unital magmas?

N. Bourbaki formally defines the free magma $M(X)$ over a set $X$. However, it does not define the free unital magma over $X$, which I am denoting by $M^{\ast}(X)$ (maybe you know some more common ...

**10**

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206 views

### Is there a non-free group $G$ whose subgroups are all freely decomposable?

Suppose that $G$ is a group such that every subgroup $H \subseteq G$ (including $G$ itself) is either free or a non-trivial free product, i.e. $H = H_1 * H_2$ with $H_1, H_2$ both non-trivial. Is ...

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**2**answers

164 views

### Why is every nilpotent-by-finite finitely generated pro-p-group always $p$-adic analytic

I'm studying the paper "On the verbal width of finitely generated pro-p groups" by Andrei Jaikin-Zapirain (link at ProjectEuclid) and I cannot see a claim made in a proof. I don't know if ...

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**1**answer

151 views

### Converse of Schreier theorem

I know that every subgroup of a free group is free (Schreier theorem).
I'm wondering that a (non-trivial) converse is true, that is, if every proper subgroup of an infinite group $G$ is free, then $G$ ...

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67 views

### Extending a representation of a free group to an extension of a mapping torus

Given a free group on $n$ generators, $F_n$, $\phi$ an automorphism of $F_n$, and a non-trivial representation $\rho: F_n \rightarrow \operatorname{Homeo}_+(\mathbb{R})$, are necessary and sufficient ...

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75 views

### Symmetric group in terms of block permutations

For $i+j+k=N$, consider the permutation $\Pi_{i,j,k}\in S_N$, which keeps the numbers $0,\ldots,i-1$ fixed, and exchanges the numbers $i,\ldots,i+j-1$ with the numbers $i+j,\ldots,i+j+k-1$.
$$\Pi_{i,j,...

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240 views

### Applications of Tits' alternative in algebraic number theory

I have recently studying Tits' alternative. The theorem statement goes like the following:
Tits' alternative: Let $G$ be any finitely generated linear group over a field. Then one of the following is ...

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**1**answer

210 views

### Are groups with the Haagerup property hyperlinear?

In his 2008 paper Hyperlinear and Sofic Groups: A Brief Guide, Pestov asked (Open Question 9.5) whether every group with the Haagerup property is hyperlinear (or sofic). Has this question been ...

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55 views

### Intersection of subgroup of a free group with the lower central series

If I have a subgroup $S$ of a free group $\mathcal{F}_m$, what can I say about the behaviour of the descending sequence of subgroups
$\left< S, \Gamma_c(\mathcal{F}_m) \right>$ (where $\Gamma_c(\...

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**2**answers

397 views

### Free groups are CT-groups [closed]

A group $G$ is called CT-group if being commutative elements is transitive relation on $G\setminus\{1\}$ i.e. if $ 1 \neq x,y,z\in G $ and $[x,y]=1, [y,z]=1 $ then $[x,z]=1$.
I encountered the fact ...

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**1**answer

120 views

### Ideal of the free Lie algebra L(x,y) generated by x

Let $L=L(x,y)$ be the free Lie algebra generated by letters $x,y.$ For a vector subspace $V\leq L$ we denote by $[V,L]$ the vector space spanned by brackets $[v,l],v\in V,l\in L.$
A vector subspace $V\...

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331 views

### Ascending chain condition for 1-element normal closures in a free group

Let $F$ be a free group of finite rank. Does $F$ satisfy the ascending chain condition on normal subgroups each of which is a normal closure of one element?
In other words, can there exist elements $...

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366 views

### Infinitely generated non-free group with all proper subgroups free

Is there any example of group $G$ satisfying the following properties?
$G$ is non-abelian, infinitely generated (i.e. it is not finitely generated) and not a free group.
$H< G$ implies that $H$ is ...

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223 views

### How can you order a free group?

A left order on a (discrete) group $G$ is a total order on $G$ satisfying $\forall g,h,k \in G: g < h \implies kg < kh$. A right order is defined symmetrically, and a biorder is an order that is ...

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279 views

### outer automorphism classification

I am trying to understand Bestvina's "A Bers-like proof of the existence of train tracks for free group automorphisms". I'm going to ask a probably trivial question ... Here we go:
The automorphism $\...

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296 views

### How does Outer Space look like without a simplex?

Considering the simplicial structure of Culler and Vogtmanns Outer Space $CV_n$. The question is now:
Let $\Delta \subset CV_n$ be a closed simplex of dimension $3n-4$ or $3n-5$, how does $CV_n \...

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598 views

### Road map to learn about $\mathrm{Out}{F_n}$

I'm a last year undergraduate student and I have taken a graduate course in geometric group theory.
I'd like to start reading some more advanced stuff in geometric group theory and in particular ...

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193 views

### Generating the monoid of injective endomorphisms of the free group

Let $F$ be the free group of rank $2$ (or any finite rank if this does not matter). The set of injective group endomorphisms $F\to F$ forms a monoid $M$ by compositions. Is there a simple looking set ...

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143 views

### Description of quasimorphisms of the free group

Let $F$ be a free group of finite rank with a fixed basis and corresponding word metric. Let $Q = Q^0_h(F, \mathbb{R})$ be the space of real homogenous quasimorphisms that vanish on the basis of $F$. ...

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66 views

### Lyndon words and free groups [closed]

It is well known that Lyndon words form a basis for free Lie algebras. Is there any analog result for free groups? What is the connection between Lyndon words and free groups? Since groups and Lie ...

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40 views

### Partially commutative elements in powers of augmentation ideal

Let $\vartheta$ a relation of parcial commutation over a set $X,$ and consider the respective free parcially commutative group $F(X, \vartheta).$ Let $K[F(X, \vartheta)]$ the parcially commutative ...

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96 views

### “Brunnian” words in solvable groups

Let $G$ be a group, and call a word $W(x_1,\dots,x_n)$ in letters $x_i$ and $x_i^{-1}$ "$G$-Brunnian" if there exist $g_1,\dots,g_n\in G$ with $W(g_1,\dots,g_n)\neq1$, but $W(h_1,\dots,h_n)=1$ as soon ...

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**1**answer

145 views

### Conjugating generators in free groups

Let $F_n := \langle x_1,\dotsc,x_n\rangle$ be the free group on $n$ generators. Let $w_1,\dotsc,w_n\in F_n$ and consider the endomorphism $\varphi:F_n\to F_n, x_i\mapsto w_ix_iw_i^-$.
I conjecture ...

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687 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,...

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746 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 ...

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101 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

217 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 ...

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**1**answer

277 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 ...

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640 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 ...

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votes

**1**answer

202 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!

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113 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 $\...

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170 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 ...

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285 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}(...

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**1**answer

430 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 ...

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**2**answers

248 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 ...

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**1**answer

358 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 ...

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**1**answer

250 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:
...

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**2**answers

406 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}\...

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**0**answers

184 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 \...

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**1**answer

312 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=\{...

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432 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

106 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

163 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

129 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.
...