Questions tagged [free-groups]
The free-groups tag has no usage guidance.
202
questions
8
votes
1
answer
470
views
Commutator length modulo finite index subgroups
We write $cl$ for the commutator length, i.e. the least number of commutators which multiply to a given element of a group.
Given an element $g$ in the commutator subgroup of the free group $G=F_2$ ...
8
votes
2
answers
625
views
A metabelian quotient of a free group
I don't know much about free groups (excepted the very basics), and the following question may be trivial, although it isn't to me.
Let $F$ be a free group with $n$ generators $x_1,\dots,x_n$. ...
8
votes
1
answer
349
views
Products of subgroups of a free group
Let $F$ be a free group, and let $A,B \leq F$ be two subgroups such that $AB$ contains a nontrivial normal subgroup of $F$. Must either $A$ or $B$ contain a nontrivial normal subgroup of $F$?
What if ...
8
votes
2
answers
294
views
Is the free abstract group residually of rank d > 2?
Let $d \geq 2$ be an integer, and let $\mathcal{F}_d$ be the family of finite groups such that $G \in \mathcal{F}_d$ if and only if every subgroup of $G$ can be generated by at most $d$ elements.
Is ...
8
votes
1
answer
304
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 ...
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
$$\max_{s\in E} |s F \mathbin{\triangle} F| \le \...
7
votes
3
answers
481
views
Membership to double cosets in free groups
Is there an elementary and efficient algorithm for testing the membership to a double coset of f.g. subgroups in a free group?
Has this membership problem been implemented in GAP/Magma?
More ...
7
votes
1
answer
463
views
$\operatorname{Out}(F_n)$ is not linear for $n > 3$
The paper The Tits alternative for $\operatorname{Out}(F_n)$ I by Bestvina, Feighn and Handel and the paper Automorphisms of free groups and Outer space by Vogtmann both state that $\operatorname{Aut}(...
7
votes
3
answers
585
views
growth of a free group automorphism is same for finite index subgroups?
Let $X=\{x_1,\dots,x_N\}$ and $F=F(X)$ be a free group generated by $X$. Let $\phi\colon F\to F$ be an automorphism of $F$. Define a growth function of $\phi$ as:
$$
\operatorname{gr}_{\phi,X}(n)=\...
7
votes
1
answer
227
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 $...
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 ...
7
votes
1
answer
202
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. $\...
7
votes
2
answers
489
views
How to solve this one-variable equation in a free group?
Let $F_n$ be a free group, $u, v \in F_n$, where $[u,v] \neq 1 $. Trying to show that the following equation does not have a solution in $F_n$ :
$$ [v,x] = [u,v] .$$
Any ideas are appreciated. I ...
7
votes
0
answers
315
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 \...
7
votes
0
answers
251
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 ...
7
votes
0
answers
194
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 ...
6
votes
4
answers
672
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 ...
6
votes
2
answers
432
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 ...
6
votes
1
answer
393
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
votes
1
answer
422
views
Do free profinite groups satisfy Howson's theorem?
Let $F$ be a free profinite group, and let $A,B \leq F$ be finitely generated closed subgroups. Must $A \cap B$ be finitely generated?
6
votes
2
answers
226
views
Finding an "optimal" quotient in a free group
Consider the abelian free group $G = \mathbf{Z}^n$ of rank $n$ and a finite subset $A \subset G \setminus \{0\}$. Since $G$ is residually finite, there is a subgroup $H \subset G$ such that $A \cap H =...
6
votes
1
answer
334
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:
...
6
votes
0
answers
125
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 $\...
6
votes
0
answers
243
views
Wild automorphisms of a free group
Let $F_X$ be a free group on a countably infinite set $X$. Let $\alpha$ be an automorphism of $F_X$ and $H$ a closed subgroup of $F_X$ in the profinite topology.
Is it possible that $\alpha(H) \...
6
votes
0
answers
363
views
Primitive elements in a free group
Let $F$ be a free group of rank $\aleph_0$ and let $H$ be a subgroup for which there exists some $a \in F$ such that $\langle H \cup \{a\} \rangle = F$. Must there be some free basis $B$ for $F$ and a ...
6
votes
0
answers
210
views
Dynamics of virtual automorphisms of free group
The setup is that $F$ is a free finitely generated group, $H, H'$ are subgroups of index $2$, and $\tau:H\to H'$ is an isomorphism.
Denote by $B_r$ the ball around $1$ of radius $r$ in $F$, in the ...
5
votes
2
answers
261
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 ...
5
votes
2
answers
253
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 ...
5
votes
2
answers
484
views
Primitive subwords in a free group of rank 2
I am not sure yet about what I exactly need to prove, but I guess I can formulate a rough statement similar to the following:
Suppose $w\in F_2$ is a primitive word whose length is big enough. Then ...
5
votes
1
answer
268
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 ...
5
votes
2
answers
427
views
Is the mapping torus of an automorphism of a free group virtually an amalgamated product?
Let $F$ be a nonabelian finitely generated free group,
let $\tau \in \mathrm{Aut}(F)$ be an element of infinite order,
and set $G = F \rtimes \mathbb{Z}$,
where the action of $\mathbb{Z}$ on $F$ is ...
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 $g$ ...
5
votes
1
answer
448
views
Normal Subgroup Growth
Let $F$ be a free group on $d$ generators.
Denote by $F_{k}$ the $k$-th term in $F$'s derived series. Put $G = F/F_k$. What is the normal subgroup growth of $G$?
Explicitly, for each natural number $...
5
votes
1
answer
389
views
finitely presented subgroup and free solvable group of class 3
Let $F(n)$ be free group of rank $n\geq 2$. Denote by $F_d(n)$ the d-th derived subgroup, that is $F_d(n)=[F_{d-1}(n),F_{d-1}(n)]$ where $F_0(n)=F(n)$. The free solvable group of rank $n$ and ...
5
votes
1
answer
277
views
Automorphism classes of the free group
As is well known, the conjugacy classes of the free group $F_2$ are parametrised by cyclically reduced words, up to cyclic permutation. In particular, it's easy to tell whether two elements of $F_2$ ...
5
votes
1
answer
277
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 ...
5
votes
1
answer
238
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 ...
5
votes
1
answer
259
views
Bases of surface groups
Let $\Gamma_g$ be a surface group of genus $g \geq 2$. A $2g$-tuple $(x_1,y_1, \dots,x_g,y_g) \in \Gamma_g^{2g}$ will be called a Surface Basis if we have the presentation $$\Gamma_g = \langle x_1, ...
5
votes
2
answers
613
views
Which polynomials are Fricke polynomials ?
Let me recall the definition which seems the most standard of Fricke polynomials.
Let $G$ be the free group with two generators $u,v$. It is not very hard to prove that there exists a unique ...
5
votes
1
answer
199
views
Converse of Schreier theorem
I know that every subgroup of a free group is free (Schreier theorem).
I'm wondering that a (non-trivial) converse is true, that is, if every proper subgroup of an infinite group $G$ is free, then $G$ ...
5
votes
0
answers
112
views
Computability of the "free envelope rank" of an endomorphism of a free group
Let $F$ be a free group freely generated by the finite set $S$ and $\sigma\colon F\to F$ be a group morphism. We define the free envelope rank of $\sigma$, written $r(\sigma)$, as the smallest $k$ for ...
5
votes
0
answers
342
views
Applications of Tits' alternative in algebraic number theory
I have recently studying Tits' alternative. The theorem statement goes like the following:
Tits' alternative: Let $G$ be any finitely generated linear group over a field. Then one of the following is ...
5
votes
0
answers
188
views
Description of quasimorphisms of the free group
Let $F$ be a free group of finite rank with a fixed basis and corresponding word metric. Let $Q = Q^0_h(F, \mathbb{R})$ be the space of real homogenous quasimorphisms that vanish on the basis of $F$. ...
5
votes
0
answers
112
views
Do the "Nielsen" IA-automorphisms of a profinite free group $\widehat{F}$ of rank 2 form a normal subgroup of $\mathrm{Aut}(\widehat{F})$?
Let $F$ be the discrete free group of rank 2, and let $\widehat{F}$ be its profinite completion, equipped with an embedding
$$i : F\hookrightarrow\widehat{F}$$
By a result of Asada, this embedding ...
5
votes
0
answers
294
views
free subgroups of $SL_2(\mathbb{Z[i]})$
The group $SL_2(\mathbb{Z})$ contains many free subgroups, for example all of the principal congruence subgroups for $n\geq 3$ and the subgroup $\left\langle \left(\begin{array}{cc}
1 & 2\\
0 &...
5
votes
0
answers
189
views
Is there a nontrivial profinite word which is trivial in any group with at most d generators?
Let $F$ be a free profinite group of rank $\aleph_0$, and let $d \in \mathbb{N}$.
Let $N_d \lhd_c F$ be the intersection of all open normal subgroups $L \lhd_o F$ for which $F/L$ can be generated by ...
5
votes
0
answers
404
views
Profinite groups, completions, and Schreier's formula
Let $G$ be a finitely generated profinite group, and $H \leq_o G$. We say that $H$ satisfies Schreier's formula in $G$ if $d(H) - 1 = (d(G)-1)[G:H]$. We say that $G$ satisfies Schreier's formula if ...
5
votes
0
answers
720
views
Conjugacy classes of elements in free groups. One-variable equations.
First of all, wasn't sure what could be a good title for this question. If mods think of a better name, pls feel free to change it...
Let $F$ be a (non-abelian) free group of finite rank, a vector $\...
4
votes
3
answers
259
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....
4
votes
1
answer
221
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!