Questions tagged [free-groups]
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206 questions
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votes
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answers
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Free subgroup of a quotient
Let $F$ be a free group on $x,y,z$. Fix $n>1$ (I am ready to assume that $n$ is large enough). Let $\mathcal{W}$ be the set of cyclically reduced words $w$ in $F$ where the letter $z$ appears at ...
- 11.3k
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
4
votes
3
answers
320
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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....
- 11.3k
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votes
0
answers
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specific qi on free groups
Let $F_n$ be the free group on $n$ generators, $n>1$.
If $\phi$ is a quasi-isometry (or a bijective bilipschitz equivalence) on $F_n$, then what can we say about the explicit form of $\phi$?
In ...
- 1,528
4
votes
2
answers
193
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Free algebras on sets of different cardinality – for what theories are they non-isomorphic?
Following the case of groups, I asked in this MSE question for a quick proof that given a free-forgetful adjunction $F\dashv U$ for some algebraic theory, we have $X\not\cong Y\implies FX\not\cong FY$....
- 10.5k
5
votes
0
answers
371
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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 &...
- 253
5
votes
2
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456
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 ...
- 11.3k
11
votes
1
answer
2k
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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
6
votes
2
answers
232
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 =...
- 373
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votes
1
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400
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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 ...
- 421
4
votes
1
answer
151
views
Genericity of irreducible automorphisms of free groups
I have seen in the literature that Irreducible outer automorphisms of a free group $F_n$ are "generic".
I would like to ask that if for example : it is true that for any generating set $X$ of $Out(...
- 339
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votes
1
answer
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Wild automorphisms of profinite groups
Is there a profinite group $G$, a continuous automorphism $\alpha$ of
$G$ and a topologically finitely generated closed subgroup $H \leq G$
such that $\alpha(H) \lneq H$ ?
Note that if an example ...
- 11.3k
2
votes
0
answers
90
views
Fully residually free groups and completion
Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?
- 11.3k
6
votes
0
answers
245
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) \...
- 11.3k
2
votes
0
answers
147
views
Rank gradient in free products amalgamating a finite subgroup
Let $A,B$ be finitely generated groups with a common finite subgroup $C$. Suppose that $[A : C] > 2, [B : C] > 1$.
Must $A *_C B$ have positive rank gradient?
See Which 3-manifolds have ...
- 11.3k
2
votes
0
answers
124
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Salvaging Howson's theorem for free profinite groups
This is an attempt to find a correct version of Do free profinite groups satisfy Howson's theorem?
Let $F$ be a free profinite group, and let $A,B \leq F$ be closed finitely generated subgroups ...
- 11.3k
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
4
votes
0
answers
172
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Example of "exotic" verbal subgroups of free groups
This will be an ambiguous question.
I am interested in various examples that appear in the literature of verbal subgroups of free groups, but which are not part of the "classical examples" like ...
- 647
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votes
2
answers
425
views
On the Magnus Representation of Free Metabelian Group
Let $F=\langle x_1,x_2\rangle$ be a free group of rank $2$ and $\Phi=F/F''=\langle \overline{x}_1, \overline{x}_2\rangle$ where $F''$ is second derived subgroup of $F$ (i.e. $F'=[F,F]$ and $F''=[F',F']...
- 427
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votes
1
answer
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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
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votes
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answers
216
views
pro-p dense subgroup in the free group
Let $G_p$ be pseudovariety of all finite $p$-group. The pro-$p$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $G_p$ is a fundamental ...
- 421
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votes
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answer
331
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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
6
votes
0
answers
369
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 ...
- 11.3k
0
votes
1
answer
193
views
Must a group of defficiency > 1 be nonabelian?
Let $F$ be a free group of finite rank $n > 2$, and let $S \subseteq F$ be a subset of cardinality at most $n-2$. Denote by $S^F$ the normal subgroup of $F$ generated by $S$. Must $F/S^F$ be ...
- 11.3k
3
votes
0
answers
117
views
Generating free groups by small subgroups and an element
Let $F$ be a free group of countably infinite rank, and let $L \leq F$ be a finite index subgroup. For some prime $p$ and $k \in \mathbb{N}$, let $\sigma \colon L \to \mathrm{GL}_k(\mathbb{Z}/p\mathbb{...
- 11.3k
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votes
2
answers
1k
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Magnus' embedding theorem
I am looking for a (preferably modern) reference to the following old result of Magnus.
Let $F$ be a free group of finite rank and
$$ F_1 = [F,F], F_2 = [F_1,F_1], \dots , F_{n+1} = [F_n,F_n], \dots ...
- 5,050
3
votes
1
answer
173
views
What is the corank of a proper char subgroup of a finite index subgroup of a free group?
Let $F$ be the free group of rank $\aleph_0$, let $L \leq F$ be a finite index subgroup, and let $M$ be a proper characteristic subgroup of $L$. Is it possible that there exists a finite subset $S \...
- 11.3k
3
votes
1
answer
145
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Thickening graphs to get honest actions
Let $X$ be a finite graph. Its fundamental group is the free group $F_n$ on (say) $n$ generators. Let further an automorphism $\phi$ of $F_n$ be given.
It is not true in general that this ...
- 11.1k
55
votes
5
answers
2k
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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
2
votes
1
answer
153
views
Collections in direct products and freeness
I am looking for references about the following type of questions:
Let $G$ and $H$ be two groups,
let $(g_i:i\in I)\subset G$ and $(h_i:i\in I)\subset H$ be collections of group elements,
and ...
user34424
8
votes
1
answer
350
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 ...
- 11.3k
23
votes
1
answer
1k
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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
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
11
votes
2
answers
778
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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
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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
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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
6
votes
1
answer
431
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?
- 11.3k
1
vote
0
answers
120
views
Free profinite products
Let $F$ be a nonabelian finitely generated free profinite group, and let $x \in F$. Must there be some $1 \neq y \in F$ such that $\langle x,y \rangle$ is isomorphic to the free profinite product of $\...
- 11.3k
4
votes
0
answers
240
views
Is a finitely generated subgroup of a free profinite group virtually a retract?
Let $F$ be a nonabelian finitely generated free profinite group, and let $H \leq F$ be a finitely generated closed subgroup. Must there be some open subgroup $H \leq U \leq F$, and a closed normal ...
- 11.3k
3
votes
0
answers
104
views
Geometric automorphism of free group respect to nonorientable suface
An outer automorphism $[ϕ]\in Out(F_n)$ is geometric if it is induced by a surface homeomorphism h:M→M, where M is a compact surface with nonempty boundary. I am wondering is it enough we only ...
- 1,598
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votes
4
answers
2k
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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
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0
answers
133
views
Equalizer in Free groups
Let $F_n$ be the free group generated by $x_1,\cdots,x_n$ and for each $1\leq i\leq n$, let a homomorphism $d_i:F_n\to F_{n-1}$ be defined as follows:
$d_i(x_r)=x_r$, if $i>r$;
$d_i(x_r)=1$, if $i=...
- 1,108
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votes
1
answer
260
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Schreier's formula and supersolvable groups
A finitely generated profinite group $G$ is said to satisfy Schreier's formula if for every open subgroup $L \leq_o G$ we have $d(L) = (d(G)-1)[G:L] + 1$. Here $d$ stands for the smallest cardinality ...
- 11.3k
1
vote
0
answers
302
views
Can a profinite completion be free pro-p?
Is there a prime number $p$ and a finitely generated residually finite group whose profinite completion is a free pro-$p$ group on a nonempty finite set?
Thanks to YCor we see that we cannot take the ...
- 11.3k
2
votes
1
answer
237
views
Conjugates and infinite index subgroups of free groups
Here I am asking for an analogue of Generating infinite index subgroups of a free group
Let $F$ be a nonabelian finitely generated free group, let $H \leq F$ be a finitely generated subgroup of ...
- 11.3k
4
votes
1
answer
515
views
Generating infinite index subgroups of a free group
Let $F$ be nonabelian finitely generated free group, let $H \leq F$ be a finitely generated subgroup of infinite index and let $x,y \in F \setminus H$. Must there be some $a \in F$ such that $[F : \...
- 11.3k
3
votes
1
answer
445
views
Co-rank of a group with $a^2b^2c^2=1$ (fundamental group of non-orientable surface)
What is the co-rank of a group $$G=\langle a_1,a_2,\dots,a_h\mid a_1^2a_2^2\dots a_h^2=1\rangle,$$ that is, finitely generated group with $h$ generators and one relation?
By co-rank, I mean the ...
- 504
3
votes
0
answers
209
views
Growth of the number of generators in hyperbolic groups
Let $G$ be an infinite hyperbolic group, and let us further assume that it is residually finite (or even LERF/GFERF) so that we have plenty of subgroups of finite index.
I would like to know if one ...
- 11.3k
3
votes
1
answer
189
views
Measuring products of finitely generated subgroups of free groups
Let $F$ be a finitely generated free group, $H_1, \dots, H_n$ finitely generated subgroups of infinite index in $F$, and $\epsilon > 0$. Must there be an epimorphism to a finite group $\phi \colon ...
- 11.3k
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