Questions tagged [free-groups]
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66 questions with no upvoted or accepted answers
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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
18
votes
0
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477
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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 $F_2$. ...
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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
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
1
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
8
votes
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186
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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 \...
- 10.1k
7
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319
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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
7
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0
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260
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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
7
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200
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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
6
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128
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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
6
votes
0
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245
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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
6
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369
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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
6
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213
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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
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113
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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 ...
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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
5
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0
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192
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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
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117
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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
5
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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 &...
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5
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189
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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 ...
- 11.3k
5
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0
answers
406
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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 ...
- 11.3k
5
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0
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737
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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 $\...
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4
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0
answers
110
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"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
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
4
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240
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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
4
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259
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Schreier's formula and descending chains
For a group $G$ we denote by $d(G)$ the cardinality of a smallest set of generators.
A finitely generated group $G$ is said to satisfy Schreier's formula if for every subgroup $H \subseteq G$ of ...
- 11.3k
3
votes
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228
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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 ...
- 181
3
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78
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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
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92
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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
3
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0
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216
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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
3
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117
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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
3
votes
0
answers
104
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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
3
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0
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209
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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
0
answers
91
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A Karrass-Solitar or Ivanov-Schupp for profinite groups
Let $F$ be a nonabelian free profinite group, $H \leq_c F$ finitely generated with $[F:H] = \infty$. Must there be some $\{1\} \neq N \lhd_c F$ such that $N \cap H = \{1\}$?
- 11.3k
3
votes
0
answers
222
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Generating set for abelianization of "mod $p$ commutator subgroup" of a free group
Let $F_n$ be a free group on $n$ letters, and fix some prime $p \geq 2$. Define
$$K_{n,p}=\text{ker}(F_n \rightarrow H_1(F_n;\mathbb{Z}/p))$$
and
$$V_{n,p} = H_1(K_{n,p};\mathbb{Q}).$$
For $x \in ...
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Test words in free profinite groups
Let $G$ be a group. An element $g \in G$ is said to be a test element if any endomorphism $\phi$ of $G$ such that $\phi(g) = g$ is an automorphism. The free group $F_2$ of rank $2$ is generated by $...
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140
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Strong converse of Kazhdan's property (T)
In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that (non-abelian) free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(...
- 1,027
2
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142
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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
2
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0
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123
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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 ...
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2
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0
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114
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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 ...
- 204
2
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44
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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
2
votes
0
answers
205
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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
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149
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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.
...
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2
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86
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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
2
votes
0
answers
90
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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
2
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0
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147
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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
2
votes
0
answers
133
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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
2
votes
0
answers
165
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Exhausting a free pro-p group
Recall that for a profinite group $G$ we define the subgroup rank to be $$\sup \{d(H): H \leq_c G\}$$ where $d(H)$ stands for the minimal cardinality of a set of topological generators of $H$.
Let $p$...
- 11.3k
2
votes
0
answers
126
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Bases for free pro-p groups
Let $p$ be a prime number, $F$ a free nonabelian finitely generated pro-$p$ group, $L \lhd_o F$ and $Y$ a basis for $L$ with $y \in Y$. Is there a basis $X$ for $F$ such that $y$ is in the abstract ...
- 11.3k
2
votes
0
answers
276
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Finite Cohomology and free groups
Let $F$ be a finitely generated nonabelian free profinite group, $p$ a prime number, $L \lhd_o F$ with $[F : L]$ coprime to $p$, $N \lhd_c^\infty F$ contained in $L$ with $L/N$ pro-$p$, and $N \leq H \...
- 11.3k