Questions tagged [free-groups]
The free-groups tag has no usage guidance.
206 questions
55
votes
5
answers
2k
views
Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?
Choose unit quaternions $q_0, q_1, q_2, q_3, q_4$ that form the vertices of a regular 4-simplex in the quaternions. Assume $q_0 = 1$. Let the other four generate a group via quaternion ...
- 22.3k
23
votes
1
answer
1k
views
products of conjugates in free groups
While trying to carry out some technical arguments in free groups, I have encountered the following problem, to which I don't know the answer.
Let $F$ be a free group and let $g,a_1,\ldots,a_n \in F$....
- 37.4k
20
votes
4
answers
2k
views
Categorical proof subgroups of free groups are free?
This is a crossport of this question from MSE.
Is there a categorical proof that subgroups of free groups are free?
How about the result that subgroups of free abelian groups are free abelian?
What ...
- 935
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
19
votes
1
answer
3k
views
What are the relations between conjugates and commutators?
The following algebraic structure came up when I was thinking about invariants of coloured knots. The elements are all elements of a noncommutative free group $F$, and the operations are:
$a^b= b^{-...
- 22.1k
18
votes
0
answers
477
views
Linear groups which don't contain products of free groups
Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The Tits alternative says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$. ...
- 17k
17
votes
3
answers
1k
views
Examples of locally hyperbolic groups
It is well-known that a subgroup of a hyperbolic group need not be hyperbolic. Let us say that a (finitely generated) group $G$ is locally hyperbolic if all its finitely generated subgroups are (...
- 383
17
votes
3
answers
974
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}...
- 25.5k
16
votes
2
answers
602
views
$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$
In a paper I found the following result:
$$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$$
However, they got the result as a corollary of a ...
- 911
16
votes
1
answer
2k
views
"Concretely" writing down elements in a free profinite group
Let $r$ be a natural number. The elements of the free group $F_r$ on $r$ generators have a nice concrete description as "words" in the $r$ generators (and their inverses). I'd like to know if there is ...
- 25.6k
16
votes
1
answer
653
views
Elements of a free group that can't be inverted by automorphisms
Let $F_n$ be a free group of rank $n$. Say that $w \in F_n$ is non-reversible if there does not exist any $f \in \text{Aut}(F_n)$ such that $f(w) = w^{-1}$.
Original Question. Intuitively, I expect ...
- 44.8k
16
votes
1
answer
918
views
Explicit path in the unitary group of a $C^*$-algebra
For $G$ a discrete group, there is a canonical inclusion $g\mapsto u_g$ of $G$ into the unitary group of the reduced $C^*$-algebra $C^*_r(G)$. Denote by $[u_g]$ the class of $u_g$ in the (topological) ...
- 11.1k
16
votes
1
answer
341
views
Must nonunit in group algebra of free group generate proper two-sided ideal?
Let $F$ be a free group and $k$ be a field. If $x$ is an element of
the group algebra $k[F]$ that is not a unit (equivalently, that is not
a nonzero scalar multiple of an element of $F$), must the 2-...
- 3,911
15
votes
1
answer
1k
views
Recognizing free groups
While I’m aware (and as it was pointed out in the comments) it is generally undecidable whether a given presentation represents a free group, I’m interested in criteria that nevertheless ensure that ...
- 1,174
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
13
votes
1
answer
548
views
Generators for the first cohomology of free groups
Let $F = \langle x_1, \dots, x_n \rangle$ be the free group on $n$ generators and $R = \mathbb Z$. The Fox derivatives $\frac{\partial}{\partial x_i} \colon F \to R[F]$ are the unique derivations ...
- 195
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
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
13
votes
1
answer
543
views
Number of trivializations of a trivial word in the free group
Let $M$ be the free monoid on $2n$ generators $x_1,X_1,...,x_n,X_n$ and consider the set $T$ of all those elements of $M$ which map to 1 of the free group on $x_1,...,x_n$ under the homomorphism $\pi$ ...
- 18.4k
13
votes
0
answers
223
views
Are there free and discrete subgroups of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$ that are not Schottky on any factor?
$\DeclareMathOperator\SL{SL}$Does there exist a free and discrete subgroup $\Gamma < \SL_2(\mathbb{R}) \times \SL_2(\mathbb{R})$ such that neither $\pi_1(\Gamma)$ nor $\pi_2(\Gamma)$ is free and ...
- 1,574
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
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
11
votes
1
answer
2k
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$?
...
- 2,303
11
votes
2
answers
778
views
History of Tarski's problems on free groups
As is known, Tarski posed his questions about first-order theories of non-abelian free groups around 1945. However, the questions were not published in his papers or books.
What is the original ...
- 893
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
11
votes
1
answer
401
views
Union of conjugates in free groups
Let $F$ be a (finitely generated) free group, $H \leq F$ of infinite index. Is it possible that $$ \bigcup_{g \in F} gHg^{-1} = F?$$
- 11.3k
11
votes
1
answer
394
views
A question on normal closures of elements in free groups.
Let $F$ be a free group of finite rank, and $p, b \in F$, where $b$ is a root element (i.e. not a proper power).
I have a case where $p^{n_k} = V_{n_k}^{-1}b^{-1} V_{n_k} \cdot U_{n_k}^{-1}b U_{n_k}$,...
- 583
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
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
10
votes
3
answers
814
views
Non-generating sets in a free group
Let $F_n$ be the free group generated by $x_i$, for $1\leq i\leq n$. Let $a_i$ be some elements of $F_n$, also for $1\leq i\leq n$. Is there a nice way to tell when the list $\{a_i^{-1}x_ia_i\}$ does ...
- 1,960
10
votes
1
answer
340
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 ...
- 337
10
votes
1
answer
555
views
Can automorphism equivalence in a free group be detected in a nilpotent quotient?
If $G$ is a group and $g_1, g_2 \in G$ let us write $g_1 \sim g_2$ if there is an automorphism $\alpha \in \operatorname{Aut}(G)$ such that $g_1^\alpha = g_2$.
Let $F = F_2$ be the free group on two ...
- 9,720
10
votes
1
answer
574
views
Length of commutators in the free group
Let $G=F_2$ be the free group of rank $2$. Is there a constant $c>0$ such that the word length $|[u,v]|$ of every commutator $[u,v]=uvu^{-1}v^{-1}$ where $u,v\in G$, $|u|,|v|>0$ is at least $c(|...
- 1,832
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
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
10
votes
2
answers
383
views
Can a positive measure subset of a free group be nowhere dense?
Let $F$ be a finitely generated free group and let $S \subseteq F$ be a subset for which there is some $\epsilon > 0$ such that for any epimorphism to a finite group $\phi \colon F \to G$ we have ...
- 11.3k
10
votes
1
answer
331
views
A subgroup of corank 1 in a free group contains a primitive element?
Let $F$ be the free group on $\{x_i\}_{i=1}^\infty$, and let $H \leq F$ be a subgroup with $\langle H \cup \{x_1\} \rangle = F$. Must there be a free basis $B$ of $F$ for which $B \cap H \neq \...
- 11.3k
9
votes
3
answers
1k
views
First-order axiomatization of free groups
Is there a way to axiomatize [non-abelian] free groups in first-order logic using the language of groups (which contains the binary operation symbol $\cdot$, and the constant symbol $e$)?
Is there ...
- 39.7k
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
9
votes
2
answers
3k
views
Number of subgroups of a given index of a free group
Given $n,d\in \mathbb{Z}^+$, how many subgroups of index $d$ does the free group of
rank $n$ have?
In case $n=1$ the question is trivial, and in case $n=2, d=2$ there are 3 such subgroups.
I think I ...
- 333
9
votes
3
answers
392
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 &...
- 111
9
votes
1
answer
377
views
Morse theory on outer space via the lengths of finitely many conjugacy classes
Let $F_n$ be the free group on letters $\{x_1,\ldots,x_n\}$ and let $X_n$ be the (reduced) outer space of rank $n$. Points of $X_n$ thus correspond to pairs $(G,\mu)$, where $G$ is a finite connected ...
- 93
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
$$G=\langle x_1,y_1,\dots,x_g,y_g \ | \ [x_1,y_1]\cdots
[x_g,...
- 1,386
9
votes
1
answer
851
views
Commutators in free groups
Let $F_m$ be the free group with $m$ generators $S:=\{x_1,\dots, x_m\}$. I am interested in the following quantity
$$
F(n):=\frac{|\{w\in [F_m,F_m]: \|w\|_S\leq n\}|}{|B_S(n)|}
$$
where $B_S(n):=\{w\...
- 2,508
9
votes
1
answer
526
views
Shortest almost trivial element of free group [duplicate]
Let $F_n$ be the free group with $n$ generators $\gamma_1,\dots,\gamma_n$.
Consider the homomorphisms $h_i\colon F_n\to F_{n-1}$ defined by adding the relation $\gamma_i=1$ in $F_n$.
What is the ...
- 45k
9
votes
1
answer
390
views
Finite presentability of semi-direct product of free group and its commutator subgroup
Let $F_n$ be a free group of rank $n \geq 2$. The group $F_n$ acts on its commutator subgroup $[F_n,\, F_n]$ by conjugation. Let $G = [F_n,\, F_n] \rtimes F_n$. It's not hard to see that $G$ is ...
- 93
9
votes
1
answer
521
views
Is a free group a product of f.g subgroups of infinite index?
Let $F$ be a free group, and let $H,K \leq F$ be finitely generated subgroups of infinite index in $F$. Is it possible that for the set of products we have $HK = F$ ?
- 11.3k
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
9
votes
1
answer
327
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. ...
- 742
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