Questions tagged [free-groups]
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206 questions
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$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$
In a paper I found the following result:
$$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$$
However, they got the result as a corollary of a ...
- 911
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1
answer
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Indicability of $\mathrm{Out}(F_n)$
A group $G$ is said to be indicable if it surjects onto $\mathbb{Z}$.
If $n=1$: $\mathrm{Out}(F_1)=\mathbb{Z}/2\mathbb{Z}$ and no finite group surjects onto an infinite group.
If $n\geq 4$: $\...
- 911
4
votes
1
answer
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Equation in the conjugacy class of a free group
I will pose the question in the form in which it originally appeared to me:
Let $a,b,c,d$ be different letters in a finite alphabet $\mathcal{Z}$. Let $Q$ and $R$ be finite words with letters from $\...
- 111
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0
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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 $...
- 355
4
votes
1
answer
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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 \...
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9
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1
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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 ...
- 45k
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1
answer
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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 ...
- 9,720
13
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1
answer
548
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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 ...
- 195
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0
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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
2
votes
1
answer
179
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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 ...
- 63
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1
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1k
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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 ...
- 1,174
4
votes
2
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412
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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]^\...
- 985
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0
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176
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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 $...
- 119
1
vote
0
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132
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
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0
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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
5
votes
1
answer
294
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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 ...
- 355
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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
9
votes
1
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377
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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
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223
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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
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320
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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 ...
- 345
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2
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269
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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
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138
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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 ...
- 369
5
votes
2
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265
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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 ...
- 1,075
2
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1
answer
151
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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 ...
- 667
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2
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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 ...
- 123
16
votes
1
answer
653
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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 ...
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3
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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 (...
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1
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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
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143
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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$?
- 1,418
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269
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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
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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 ...
- 211
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1
answer
224
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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 ...
5
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2
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228
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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$ ...
- 1,996
3
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1
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297
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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(...
- 517
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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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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 ...
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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 ...
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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
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1
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425
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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
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1
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127
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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 \...
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1
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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(|...
- 1,832
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2
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438
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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 ...
- 5,349
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0
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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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votes
3
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523
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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 ...
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votes
1
answer
219
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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
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1
answer
219
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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. $\...
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vote
0
answers
44
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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{...
- 302
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0
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142
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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 ...
- 353
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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
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2
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257
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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 ...
- 329