Questions tagged [free-groups]

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38 views

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 ...
4
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0answers
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 ...
7
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3answers
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 ...
2
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1answer
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 ...
7
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1answer
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. $\...
1
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0answers
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{...
1
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0answers
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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1answer
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 ...
4
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2answers
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 ...
5
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1answer
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 $G$ ...
3
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0answers
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 ...
3
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0answers
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,...
5
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0answers
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 ...
9
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1answer
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 ...
0
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0answers
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(\...
5
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2answers
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 ...
1
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1answer
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\...
11
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0answers
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 $...
19
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0answers
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 ...
5
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0answers
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 ...
4
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1answer
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 $\...
7
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0answers
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 \...
10
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2answers
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 ...
5
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0answers
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 ...
5
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0answers
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 $F$. ...
1
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0answers
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 ...
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
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 ...
4
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1answer
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 ...
9
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2answers
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,...
12
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1answer
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 ...
2
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0answers
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
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1answer
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 ...
13
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1answer
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 ...
6
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4answers
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 ...
4
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1answer
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!
6
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0answers
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 $\...
5
votes
1answer
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 ...
6
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1answer
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}(...
9
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1answer
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 ...
8
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2answers
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 ...
6
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1answer
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 ...
6
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1answer
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: ...
8
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2answers
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}\...
2
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0answers
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 \...
12
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1answer
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=\{...
7
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3answers
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
1answer
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
1answer
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
0answers
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. ...