Questions tagged [free-groups]

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36 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(\...
5
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2answers
390 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 ...
1
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1answer
107 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\...
8
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0answers
234 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 ...
18
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0answers
280 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 ...
4
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0answers
188 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 ...
4
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1answer
255 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 $\...
7
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0answers
289 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 \...
10
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2answers
507 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 ...
5
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0answers
165 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 ...
5
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0answers
132 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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0answers
63 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 ...
2
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0answers
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 ...
4
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0answers
91 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 ...
4
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1answer
136 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 ...
9
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2answers
609 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
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1answer
558 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
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0answers
99 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
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1answer
205 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 ...
13
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1answer
247 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
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4answers
604 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
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1answer
199 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
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0answers
107 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
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1answer
135 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 ...
6
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1answer
262 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
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1answer
412 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
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2answers
247 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
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1answer
345 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 ...
6
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1answer
237 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
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2answers
379 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
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0answers
177 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 \...
12
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1answer
276 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
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3answers
378 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
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1answer
103 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
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1answer
143 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
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0answers
126 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
1answer
263 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
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2answers
367 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
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0answers
155 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
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0answers
79 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....
5
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0answers
99 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 ...
10
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1answer
559 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
2answers
349 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
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2answers
915 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
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1answer
92 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
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2answers
227 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
2answers
744 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
2answers
172 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
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0answers
101 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 ...
4
votes
2answers
145 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 $X\not\cong Y\implies FX\not\cong FY$....