Questions tagged [free-groups]
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206 questions
9
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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
8
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2
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295
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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 ...
- 11.3k
5
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0
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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
2
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0
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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
1
vote
1
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442
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Action of the pure braid group on the commutator subgroup of a free group
Let $P=P_n$ be the pure braid group on $n$ strands and $F=F_n$ the free group on $n$ generators. I'm interested in a nice description of the action of $P$ on the derived subgroup $F'$ which somehow ...
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1
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0
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275
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Explicitly showing that a free group is LERF [closed]
Let $F$ be a free group on a finite set $X$, and let $M$ be a finitely generated subgroup.
Marshall Hall's theorem states that $M$ is closed in the profinite topology on $F$. That is, $M$ is the ...
- 11.3k
3
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0
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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
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0
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421
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Marshall Hall's theorem for surface groups [closed]
Let $\Gamma_g$ be a surface group of genus $g \geq 2$, that is we have a presentation: $$\Gamma_g = \langle x_1,y_1 \dots, x_g,y_g \vert \prod_{i = 1}^g [x_i,y_i] = 1\rangle$$
Let $H \leq \Gamma_g$ ...
- 11.3k
2
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1
answer
306
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An epimorphism into a profinite group
Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, $L \lhd_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose that there is a continuous surjection from $G$ onto a free ...
- 11.3k
2
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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
1
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0
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Dense free subgroups
Let $F$ be a free pro-$p$ group (for a prime number $p$) on a finite set $X$, $\Phi$ the abstract subgroup generated by $X$, $\{1\} \neq N \lhd_c F$. Is it possible that $\Phi \cap N = \{1\}$?
- 11.3k
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A bound on the size of the center
Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$
- 11.3k
2
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0
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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
4
votes
2
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337
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A Karrass-Solitar theorem for surface groups
Let $\Gamma_g$ be a surface group of genus $g \geq 2$. That is, there is a presentation $$\Gamma_g = \langle x_1, y_1, \dots, x_g, y_g \vert \prod_{i = 1}^{g}[x_i,y_i] = 1\rangle$$
Is there a ...
- 11.3k
5
votes
1
answer
264
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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 $$\Gamma_g = \langle x_1, ...
- 11.3k
2
votes
0
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202
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Accessible subgroups of free groups
Let $F$ be a nonabelian free finitely generated group, and $F = G_0 \rhd G_1 \rhd G_2 \dots$ a strictly descending subnormal chain of subgroups ($G_n \lhd G_{n-1}$ for each $n \in \mathbb{N}$) each ...
- 11.3k
11
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1
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401
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Union of conjugates in free groups
Let $F$ be a (finitely generated) free group, $H \leq F$ of infinite index. Is it possible that $$ \bigcup_{g \in F} gHg^{-1} = F?$$
- 11.3k
2
votes
1
answer
286
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Can a closure make the index finite?
Let $F$ be a free finitely generated group, $H \leq F$ of infinite index. Let $c : F \rightarrow \hat{F}$ be the embedding in the profinite completion. Denote by $\tilde{F}, \tilde{H}$ the closure of $...
- 11.3k
0
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0
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110
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Bases and transversals
Let $F$ be a free finitely generated group, $L \leq H \leq F$ subgroups of finite index.
Given bases $B$ of $F$ and $C$ of $H$, must there be a Schreier transversal $T$ for $H$ in $F$ such that the ...
- 11.3k
2
votes
1
answer
286
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Making a profinite group free
Let $F$ be a free profinite group, $G$ a profinite group. Suppose that the free profinite product $F \amalg G$ is a free profinite group. Must $G$ be a free profinite group?
For abstract groups the ...
- 11.3k
3
votes
1
answer
739
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Bases of free groups
Let $F$ be a free group on a finite set $X$. Let $A \subseteq X$ be a subset of $X$ contained in some $H \leq F$, a subgroup of finite index in $F$. Must there be a basis (free generating set) for $H$ ...
- 11.3k
4
votes
1
answer
455
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Generators of Sylow subgroups
Is there a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for each finite supersolvable group $G$, and a Sylow subgroup $S \leq G$ we have $d(S) \leq f(d(G))$?
Here $d(H)$ denotes the ...
- 11.3k
5
votes
0
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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
4
votes
0
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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
2
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366
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(Quadratic) equation in free group?
I know almost nothing about this field, so my following question may be stupid.
We know in a free group F, if $ax^{-1}a^{-1}x=1$, then $x=t^{k}, a=t^{l}$ for some t.
Now consider a more general ...
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2
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0
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126
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Free profinite completions
Let $m,n \in \mathbb{N}$. Which residually finite groups $G$ generated by $m$ elements, have the free profinite group on $n$ generators as their profinite completion?
- 11.3k
5
votes
1
answer
452
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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 $...
- 11.3k
4
votes
2
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861
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Schreier's index formula
A finitely generated group G is said to satisfy Schreier's index formula if for every subgroup H of index k in G we have: d(H) - 1 = k(d(G) - 1). For example, a finitely generated free group satisfies ...
- 11.3k
3
votes
1
answer
190
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Intersection growth of free profinite groups
Let $F_n$ be a free profinite group of finite rank $n$ and let $V_k$ denote the intersection of all open subgroups of $F_n$ of rank at most $k$ ($k \in \mathbb{N}$).
My questions are:
Can I ...
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0
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1
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446
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finite index, self-normalizing subgroup of $F_2$ [closed]
Denote $F_2=\langle a, b\rangle$ to be the free group on two generators $a, b$.
Let $H\leq F_2$ to be a subgroup with finite index $n$, so $H\cong F_{n+1}$ by Nielsen–Schreier theorem, recall that $...
- 1,528
9
votes
2
answers
3k
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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 ...
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16
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1
answer
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"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
6
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0
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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 ...
4
votes
3
answers
394
views
Algebraically Independent Numbers and Affine Linear Maps
Suppose I have two lists $\alpha_1,\dots,\alpha_k$ and $\beta_1,\dots,\beta_k$ of real numbers such that all $2k$ numbers are mutually algebraically-independent over the rationals. For each $i \in \{1,...
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2
votes
2
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274
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Generating certain subgroups of free groups via primitive elements
Let $F_n$ be the free group on $n\geq 2$ generators and let $H < F_n$ be a finite index normal subgroup. Let $P\subset F$ be the subset consisting of primitive elements (an element of $F_n$ is ...
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3
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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
2
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0
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136
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Elements of minimal length in normal closures of elements in free groups
Let $F_n$ be a free group of rank $n$. Let $w\in F_n$ be cyclically reduced.
What can be said about the element(s) of minimal length from the $\textit{ncl}(w)$ (normal closure of $w$ in $F_n$)? Under ...
- 583
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0
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603
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Stable commutator length of elements in free groups.
http://arxiv.org/pdf/math/0611889v4.pdf (page 13)
In the above paper by Danny Calegari he says that the result $\text{scl}(g) \geq 1/2$ (i.e. a stable commutator length $\text{scl}(g) := \...
- 583
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votes
1
answer
286
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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$ ...
- 9,720
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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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1
answer
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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
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2
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490
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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] .$$
Any ideas are appreciated. I ...
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1
answer
851
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Commutators in free groups
Let $F_m$ be the free group with $m$ generators $S:=\{x_1,\dots, x_m\}$. I am interested in the following quantity
$$
F(n):=\frac{|\{w\in [F_m,F_m]: \|w\|_S\leq n\}|}{|B_S(n)|}
$$
where $B_S(n):=\{w\...
- 2,508
8
votes
1
answer
481
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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 $G=F_2$ ...
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16
votes
1
answer
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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
3
votes
1
answer
922
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Direct proof that a group is Hopfian
Consider the group $G=\langle x_1,x_2,x_3|x_1^2,x_2^2,x_3^2\rangle$. Using a slightly modified version of S. Ivanov's proof here that free groups are residually finite, I can show that this group is ...
- 881
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votes
3
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814
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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
19
votes
1
answer
3k
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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
3
votes
1
answer
3k
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Normal Subgroups of Free Products
Let $G=A\ast \mathbb{Z}$ be the free product of a group $A$ and the cyclic group $\mathbb{Z}$ and suppose $K$ is a subgroup of $G$. By Kurosh Subgroup Theorem we know that $K=F\ast (\ast_{i\in I}(K\...
- 2,233
5
votes
2
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530
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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 ...
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