Questions tagged [free-groups]
The free-groups tag has no usage guidance.
206 questions
5
votes
1
answer
232
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 $G$ ...
- 125
3
votes
0
answers
78
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 ...
- 384
3
votes
0
answers
92
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,...
- 3,001
5
votes
0
answers
360
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 ...
- 302
11
votes
1
answer
381
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 ...
- 1,027
0
votes
0
answers
132
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(\...
- 31
6
votes
2
answers
442
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 ...
- 61
1
vote
1
answer
297
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\...
- 1,484
11
votes
0
answers
382
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 $...
- 3,181
20
votes
0
answers
625
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 ...
- 599
10
votes
1
answer
821
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 ...
- 6,652
4
votes
1
answer
338
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 $\...
- 257
7
votes
0
answers
319
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 \...
- 255
12
votes
3
answers
1k
views
Road map to learn about $\operatorname{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 about ...
- 257
7
votes
0
answers
260
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 ...
- 121
5
votes
0
answers
192
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$. ...
- 435
1
vote
0
answers
76
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 ...
- 1,269
2
votes
0
answers
44
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 ...
- 121
4
votes
0
answers
110
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 ...
- 2,519
4
votes
1
answer
201
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 ...
- 1,623
9
votes
2
answers
939
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,...
- 1,386
13
votes
1
answer
1k
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 ...
- 885
2
votes
1
answer
219
views
Projective G-group
Let $G$ be a fixed group. By definition, a $G$-group is a group $X$ with a $G$-action that respects the group operation of $X$. A free $G$-group means a group freely generated by a free $G$-set. A &...
- 2,178
5
votes
1
answer
272
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 ...
- 2,178
15
votes
1
answer
413
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 ...
- 5,349
6
votes
4
answers
676
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 ...
- 4,164
4
votes
1
answer
233
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!
- 43
6
votes
0
answers
128
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 $\...
- 52.6k
5
votes
1
answer
255
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 ...
- 53
7
votes
1
answer
523
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}(...
- 275
10
votes
1
answer
534
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 ...
- 275
8
votes
2
answers
269
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 ...
- 153
6
votes
1
answer
405
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 ...
- 193
6
votes
1
answer
366
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:
...
- 489
8
votes
2
answers
522
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}\...
- 125
2
votes
0
answers
205
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 \...
- 21
12
votes
1
answer
415
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=\{...
- 949
7
votes
3
answers
620
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:
$$
\operatorname{gr}_{\phi,X}(n)=\...
- 1,050
3
votes
1
answer
133
views
Maximal power in a sequence of iterated commutators in the rank two free group
I have the following problem: in the free group $F_2=\langle a,b\rangle$, we define the sequence
$\begin{cases}
w_0=a, \\
w_1=b, \\
w_{n+2}=[w_{n+1},w_{n}] & \text{for }n\ge 0.
\end{cases}$
So $...
- 1,317
9
votes
1
answer
193
views
Detecting/Characterising positive elements in free groups
Let $X$ be a set, and let $F(X)$ be the free group generated by $X$.
I will say that an element of $F(X)$ is positive if it is in the monoid generated by all the conjugates in $F(X)$ of every member ...
- 2,178
2
votes
0
answers
149
views
Concentration of Reduced words
This might be a rather broad question, and I'll be satisfied with some intuition and pointers to relevant literature. However, I'll certainly fill in more context and details based on any feedback.
...
- 949
4
votes
1
answer
447
views
Finite index subgroups of a RAAG
Let $G$ be the group given by the presentation
$$\langle x,y,z,w \ | \ xy = yx, yz = zy, zw = wz\rangle.$$
This is a right-angled Artin group (RAAG) whose graph is a path on $4$ vertices.
We can ...
- 11.3k
3
votes
2
answers
700
views
Subgroup of a free group that is characteristic but not totally characteristic
Looking for a counter example (if it exists) and a reference for further reading. Can there be a subgroup of finite index in a finitely generated free group that is characteristic but not totally ...
- 193
7
votes
0
answers
200
views
Structure constants of Lyndon-Shirshov basis of the free Lie ring
Let $X$ be an alphabet, ${\sf Lyn}$ be the set of Lyndon words on $X$ and $L$ be the free Lie ring on $X.$ For $w\in {\sf Lyn}$ we denote by $[w]$ the corresponding element of the Lyndon-Shirshov ...
- 1,484
2
votes
0
answers
86
views
Automorphisms of a free topological product
Let $G$, $G_1$, $G_2$ be Hausdorff topological groups. I am mainly interested in the case when those groups are profinite.
Let $G$ act continuously on $G_1$ and $G_2$ via continuous automorphisms, i.e....
- 381
5
votes
0
answers
117
views
Do the "Nielsen" IA-automorphisms of a profinite free group $\widehat{F}$ of rank 2 form a normal subgroup of $\mathrm{Aut}(\widehat{F})$?
Let $F$ be the discrete free group of rank 2, and let $\widehat{F}$ be its profinite completion, equipped with an embedding
$$i : F\hookrightarrow\widehat{F}$$
By a result of Asada, this embedding ...
- 10.7k
11
votes
1
answer
964
views
Normal closures of finitely generated subgroups of a free group
Is it true that for every finitelty generated subgroup $H$ of infinite index in a free
group $F$ on the two letters $\{x,y\}$, there exists a finite index
subgroup $K$ of $H$, such that the normal ...
- 11.3k
13
votes
2
answers
515
views
Free groups and free restricted Lie algebras
If $G$ is any group and $\gamma_k(G)$ denotes the $k$th term in the lower central series of $G$, then the commutator bracket on $G$ endows
$$\mathcal{L}(G) = \bigoplus_{k=1}^{\infty} \gamma_k(G) / \...
- 44.8k
9
votes
2
answers
2k
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,
...
- 923
1
vote
1
answer
101
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 ...
- 348