Questions tagged [free-groups]
The free-groups tag has no usage guidance.
202
questions
4
votes
1
answer
293
views
Extending primitive systems in free groups
It is a version of a question I asked on Math.StackExchange (no answers yet). Suppose $a_1,a_2,\ldots,a_n,b$ is a primitive system in a non-abelian free group $F$ (a part of a basis of $F$). Let $c \...
- 41
9
votes
1
answer
508
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 ...
- 43.7k
10
votes
1
answer
549
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 ...
- 9,195
12
votes
1
answer
516
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 ...
- 155
3
votes
0
answers
218
views
What is known about the map $\text{Mod}_g^1 \rightarrow \text{Aut}(F_{2g})$?
Follow up question, edited in on 12/20 below:
Letting $\text{Mod}_g^1$ be the mapping class group of a surface with one boundary component (and basepoint on the boundary) and identify its fundamental ...
- 49
2
votes
1
answer
154
views
Almost free group without the Specker group as a subgroup
An Abelian group is almost free whenever every countable subgroup is free Abelian. Famously, the Specker group $\mathbf Z^{\mathbf N}$ is almost free. What are examples of almost free groups that are ...
- 21
14
votes
1
answer
988
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 ...
- 1,175
3
votes
2
answers
287
views
Units of the group algebra of a free group
Let $K$ be a field of characteristic zero and $F_n$ be a free group of rank $n$. What is known about the group of units $K[F_n]^\times$?
In the case of $n=1$, there are only trivial units: $K[F_1]^\...
- 401
0
votes
0
answers
115
views
Order of elements in amalgamated free products
Reading the book "A Course in the Theory of Groups" by D. J. S. Robinson, I was looking at the proof of 6.4.3 (iii), which states (suppose we are in the case of two groups): if $G_1$ and $...
- 109
1
vote
0
answers
126
views
A generalisation of residual finiteness?
A group $\Gamma$ is Residually Finite (RF) if
$\forall g \neq e \in \Gamma$ there is a homomorphism $h: \Gamma \to G$ where $G$ is a finite group such that $h(g) \neq e$. Free groups are known to be ...
- 305
2
votes
0
answers
133
views
Strong converse of Kazhdan's property (T)
In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that (non-abelian) free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(...
- 1,027
5
votes
1
answer
283
views
Words which are not inverted by any endomorphism
Let $w$ be a word in a free group $F_2$ of two generators $x_1, x_2$ such that there does not exist any endomorphism of free group which takes $w$ to $w^{-1}$. Let $w_1, w_2$ be two words in the same ...
- 305
2
votes
0
answers
120
views
Further questions to limit groups and an article of Fujiwara and Sela
I already have asked a question to the following article:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, ...
- 309
9
votes
1
answer
367
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
13
votes
0
answers
199
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 ...
- 1,364
8
votes
1
answer
305
views
If G is a finitely generated group with vcd(G) finite, is vcd(H) finite for H, where H is an automorphism group of G?
$\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Out{Out}$Here I mean $\vcd(G)$ to be the virtual cohomological dimension of $G$. Some ...
- 335
7
votes
2
answers
254
views
Surjections from genus $n$ surface group to free group of rank $n$
Let $\Sigma_n$ be a genus $n$ surface, let $\mathcal{H}_n$ be a genus $n$ handle body, and let $F_n$ be a free group of rank $n$. Fix an identification of $\pi_1(\mathcal{H}_n)$ with $F_n$. I know ...
- 73
0
votes
0
answers
110
views
Fundamental domain of Möbius transformations
In the book Indra's pearls, Möbius transformations are used to construct Kleinian fractals, which are limit sets of a free group generated by two Möbius transformations $a$ and $b$.
In the process of ...
- 343
5
votes
2
answers
254
views
Stable equivalence of generating sets of a finitely-generated group?
This question came about when I was naively considering to what extent generating sets for finitely-generated groups are unique.
Let $G$ be a group and let $\phi_1, \phi_2 : F_k \to G$ be two ...
- 1,065
2
votes
1
answer
140
views
Nontrivial abelianization of torsion-free pro-$p$-group which contains a dense free subgroup is infinite?
Let $G$ be a topologically finitely generated pro-$p$-group. Assume that $G$ is torsion-free, it contains a dense free subgroup of infinite rank and the (topological) abelianization $G^{\text{ab}}$ of ...
- 491
4
votes
2
answers
524
views
Length of a product of conjugates of an element in a free group
Let $G$ be a free group generated by a set $S$. For $g\in
G$, let $l(g)$ be the length of $g$ with respect to $S$.
Now for $a\in G$ and $g_1,\dotsc,g_n\in G$, let $$T=g_1^{-1}ag_1g_2^{-1}ag_2\dotsm ...
- 125
16
votes
1
answer
640
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 ...
- 43.6k
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
7
votes
1
answer
238
views
Primitive elements in a free group with trivial projection
For a free group $F$, an element $w$ is primitive if it is part of some free basis for $F$.
Let $\pi:F[x_0,x_1,...,x_n]\rightarrow F[x_1,x_2,...,x_n]$ be defined $\pi (x_0)=1$ and $\pi (x_i)=x_i$ for $...
3
votes
1
answer
136
views
Is is true that a proper subword cannot lie in the normal closure of a word?
Let $F$ be a free group and $w\in F$ a cyclically reduced word. Let $v$ be a non-trivial proper subword of $w$. Is it true that $v\notin \langle w^F\rangle$?
- 1,378
3
votes
3
answers
249
views
Perfect group that is split extension of a normal free subgroup of finite index
Does there exists a non-trivial free group $F$ and a finite group $L$ acting on $F$ such that the semidirect product $F\rtimes L$ is perfect?
Thanks @YCor for reformulating the question.
- 585
5
votes
2
answers
264
views
If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, must the fixed points of $1_\mathbb{Z}$ be a finitely generated group?
I've copied over this question from what I asked on StackExchange, in the hope that an expert here can readily answer the question.
Is there an example of a group $G=K\rtimes \mathbb{Z}$ satisfying ...
- 171
2
votes
1
answer
196
views
Compute reduced word for element in particular free group
Let's consider some particular free group, for instance, the example $G = \langle A, B \rangle < \text{SL}(2, \mathbb{Z})$ in Wikipedia about ping-pong lemma, and some element $g \in G$.
Problem #1 ...
- 105
5
votes
2
answers
224
views
Bound on the period of the identity (in a free group) for an automorphism followed by left-multiplication
Let $F$ be a finite-rank free group, $g$ an element of $F$ and $\Phi\colon F \to F$ an automorphism. Consider the dynamical system $\psi_g\colon F \to F$ defined by $x \mapsto g\Phi(x)$. Say that $g$ ...
- 1,986
3
votes
1
answer
229
views
Injectivity of certain homomorphisms on free groups
Consider free groups $F(A)$ and $F(B)$ on finite generating sets $A,B$. Write $A$ and $B$ as the disjoint unions $A=A_1\sqcup A_2$ and $B=B_1\sqcup B_2$. We consider the free groups $F(A_i)$ and $F(...
- 487
2
votes
0
answers
123
views
Probability of a finite cylinder set in a free group
Let $\mathbb{F}_n$ be the free group (each elemen is in its reduced form) generated by the set $\Sigma_n = \{a_1, a_2, \cdots, a_n, a_1^{-1}, a_2^{-1}, \cdots, a_n^{-1}\}$ and let $e$ denote the ...
- 235
9
votes
1
answer
357
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
2
votes
0
answers
104
views
understanding the definition of subgroup of the Grothendieck-Teichmuller group
Definition. Let $\widehat{G T}^{1}$ be the set of elements $f$ in the derived subgroup $\hat{F}_{2}^{\prime}$ of $\hat{F}_{2}$ such that $x \mapsto x$ and $y \mapsto f^{-1} y f$ extends to an ...
- 184
8
votes
0
answers
163
views
Uniform amenability at infinity
Let's recall that a group $G$ is amenable if for any finite subset $E\subset G$ and any $\epsilon>0$
there is a finite subset $F\subset G$ such that
$$\max_{s\in E} |s F \mathbin{\triangle} F| \le \...
- 9,631
0
votes
1
answer
345
views
Are all Coxeter groups virtually free or virtually surface groups?
From Surface subgroups of Coxeter and Artin groups (Gordon, Long and Reid, 2003) DOI link, we can read that (Theorem 1.1) a Coxeter group is either virtually free or contains a surface group ($\pi_1$ ...
- 563
1
vote
1
answer
124
views
Free partners in semi-direct products
Let $G = N \rtimes K$ be a semi-direct product of groups and suppose that $K$ is a finite group. Call the set $\mathcal{F} = \{ \alpha \in G \mid \langle \alpha, K \rangle = \langle \alpha \rangle \...
10
votes
1
answer
542
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(|...
- 1,802
2
votes
2
answers
348
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 ...
- 5,319
5
votes
0
answers
112
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 ...
- 151
7
votes
3
answers
483
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 ...
- 3,171
2
votes
1
answer
214
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 ...
- 33
7
votes
1
answer
204
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
vote
0
answers
43
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{...
- 250
1
vote
0
answers
134
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 ...
- 343
10
votes
1
answer
302
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
4
votes
2
answers
248
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 ...
- 289
5
votes
1
answer
207
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
77
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 ...
- 364
3
votes
0
answers
89
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,...
- 2,901
5
votes
0
answers
344
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 ...
- 271