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10
votes
3answers
440 views

Example of reflective subcategory of (Groups) whose reflector doesn't preserve finite products

A subcategory $D$ of a category $C$ is called reflective, if the embedding $D \hookrightarrow C$ has a left adjoint $L:C \to D$. The left adjoint $L$ is called the reflector. If the category $C$ is ...
2
votes
0answers
50 views

Pro-V topology on a free group

Let $W=G_p*G_q$, where $G_p$ and $G_q$ are pseudovarieties of all finite $p$-groups and all finite $q$-groups respectively, with $p$ and $q$ fixed prime numbers. The pro-$W$ topology on a group $G$ is ...
3
votes
1answer
141 views

Are braid groups conjugacy separable?

I would like to re-ask a question that was raised in the comments here: Normal subgroups of braid groups Namely, a group $G$ is called conjugacy separable, if it holds that for every two elements $x,...
5
votes
0answers
79 views

Random pro-p groups via iterated uniformly random central extensions

Inspired by this question on math.se, I want to understand the following construction of a random pro-$p$ group: We want to construct an inverse system $$\cdots \xrightarrow{\alpha_i} G_i \...
1
vote
1answer
131 views

Some question on haar measure for sumsets of closed subsets of profinite groups

Let $H$ be a profinite group with the haar measure $\mu_H$. Let $H_1$ and $H_2$ be closed subgroups of $H$. $H_1$ and $H_2$ have their own haar measures $\mu_{H_1}$ and $\mu_{H_2}$ respectively. ...
10
votes
0answers
295 views

A question concerning model theory of groups

Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not ...
0
votes
2answers
527 views

Is the absolute Galois group the same as the automorphism group? [closed]

Is the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})$ the same as the group $\mathrm{Aut}_{\mathbf{Q}}(\overline{\mathbf{Q}})$ the automorphism group in the category of $\...
0
votes
1answer
90 views

Uniform pro-p groups as a semi-direct product

Let $G$ a finitely generated uniform pro-$p$ group. Then $G/[G,G]$ is abelian and so it is of the form $\mathbb{Z}_p^r\times T$ for some integer $r$ and finite $p$-group $T$. Therefore, $[G,G]$ is ...
8
votes
1answer
192 views

Finite groups: equations with many solutions

Let $\omega$ be a word in the free group on generators $x_1,x_2,\ldots,x_n,g_1,g_2,\ldots,g_k$, where $n>0$ and $k\geq 0$. For any finite group $G$ and elements $g_1,g_2,\ldots g_k$ in $G$ we ...
9
votes
1answer
431 views

Dessins d'enfants and absolute Galois group

I would like to know what is the recent progress about the group homomorphism $$ \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F_{2}})$$ $\mathrm{Gal}(\overline{\mathbf{...
3
votes
1answer
121 views

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 ...
1
vote
0answers
56 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?
5
votes
0answers
173 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) \...
1
vote
0answers
82 views

Profinite rank of Fuchsian groups

Let $G$ be a Fuchsian group whose profinite completion is finitely generated. Must $G$ be finitely generated? A Fuchsian group is a discrete subgroup of $\mathrm{SL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(...
3
votes
2answers
334 views

Subgroups of hyperbolic groups

Let $G$ be a finitely generated hyperbolic group, and let $H \leq G$ be a subgroup whose profinite completion is finitely generated. Must $H$ be finitely generated? In view of Ian Agol's answer, I ...
2
votes
1answer
129 views

Is the Frattini subgroup of a free profinite group trivial?

Let $\widehat{F_2}$ be the free profinite group of rank 2. Is its Frattini subgroup $\Phi(\widehat{F_2})$ trivial? I know that the Frattini subgroup of a pro-$p$ group is open. On the other hand, the ...
5
votes
1answer
228 views

abelian and nonabelian parts of Aut($\widehat{F_2}$)

Let $F$ be the free profinite group on two generators. Let $\text{IA}(F) := \ker\left(\text{Aut}(F)\rightarrow GL_2(\widehat{\mathbb{Z}})\right)$, the group of "IA automorphisms" of $F$. (I'm also ...
0
votes
0answers
180 views

Basic question about power series and complete group algebras

This is a pretty basic question, but I suspect it might be too exotic for math.stackexchange. Let $\mathbb{Z}_p$ be the $p$-adic integers. For free pro-$p$ group $F_r$ of rank $r$, we can consider ...
1
vote
0answers
70 views

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 ...
6
votes
1answer
276 views

Inverse limit of $\Bbb Q/q\Bbb Z$ isomorphic to finite adeles?

Let $\Bbb Q/q\Bbb Z$, for some positive rational $q$, denote the quotient group of the discrete rationals by the subgroup of integers times $q$. For any $q_1, q_2 \in \Bbb Q^+$ and $n \in \Bbb N^+$ ...
3
votes
0answers
97 views

Profinite topology on free metabelian group

Let $M$ be free metabelian group of rank $n$. By work of Coulbois, M is $LERF$ and is not $RZ_2$, that means every finitely generated subgroup of $M$ is closed in the profinite topology of $M$ but the ...
2
votes
1answer
184 views

Profinite groups, directed sets and $H^1$

Usually whenever one reads the definition of profinite group, one starts with an ordered set $I$ which is directed, meaning that for every $i,j\in I$ there is some $k\in I$ such that $i\leq k$ and $j\...
12
votes
1answer
315 views

Applications of Lubotzky's linearity theorem?

Lubotzky's theorem is a necessary and sufficient set of conditions for a finitely generated discrete group to be linear, i.e. isomorphic to a subgroup of $GL_n(K)$, where $K$ is a field of ...
1
vote
0answers
59 views

dense subgroup in pro v topology

Let $V$ be a extension closed pseudovariety. The pro-$V$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $V$ is a fundamental system of ...
1
vote
0answers
47 views

Independence of inverse system to define continuous cohomology for profinite groups

I have a problem concerning cohomology of profinite groups as it is defined e.g. in Gille's and Szamuely's "Central Simple Algebras and Galois Cohomology" on page 86. For a profinite group $G=\...
1
vote
0answers
99 views

A discrete presentation for a free prop-$p$-group

Let $n\geq 1$ be an integer and $p$ a prime. Suppose that $\mathcal{F}(n,p)$ is the free prop-p-group of rank $n$. Question: For each pair $(n,p)$, is it known a discrete free group $\mathfrak{F}$ ...
3
votes
0answers
184 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 ...
13
votes
1answer
485 views

Avoiding countable subgroups of a group homeomorphic to the Cantor space

The following question is motivated by the paper [Brian, Mislove, Every compact group can have a non-measurable subgroup]. A positive solution to a variation of the following problem implies a ...
2
votes
0answers
101 views

pro-p topology on a free group

James Howie in the paper "The p-adic topology on a free group:a counterexample" showed that in the free group $F$ generated by $x$ and $y$,if $a=xy^2$, $b_1=x^{-2}y^{-3}$ and $b_2=x^{-2}(xy)^5$, then ...
7
votes
2answers
572 views

Avoiding countable subgroups of general uncountable groups

The following problem is a general form of another problem (motivation is available there). Initially, the problems were posted together, but the first one is solved below, a solution that does not ...
2
votes
1answer
201 views

Finite quotients of an infinite product of finite groups

Let $G$ be a finite group. Consider the direct product $\Gamma=\prod_{i=1}^{\infty}G$ of (countably) infinitely many copies of $G$. For every finite set of numbers $\{i_1,\ldots,i_n\}$ we have the ...
0
votes
0answers
114 views

Subgroups of powers of the alternating group on 5 elements

Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H \...
1
vote
0answers
95 views

Monadicity of profinite algebras

We can show that the category of profinite algebras, cofiltered limits of finite algebras, is monadic over Stone spaces as follows. So, I wonder if there are any other examples. In case that I was ...
0
votes
2answers
144 views

Is the Frattini subgroup of a f.g virtually pro-p group open?

Let $G$ be a finitely generated profinite group, and $p$ a prime number. Suppose that there exists some open pro-$p$ subgroup $H \leq_o G$. Must $G$ have only finitely many maximal open subgroups? ...
3
votes
2answers
250 views

A galois group is topologically finitely generated or finitely generated as a $\mathbb{Z}_p$-module

The basic set up is the following: Let $K$ be a number field. let $p$ be an odd prime. Let $\Sigma_p$ be the set of primes of $K$ lying above $p$. Let $M$ be the composite of all finite $p$-extensions ...
1
vote
0answers
82 views

Dense subgroups in subgroups of profinite groups

Let $G$ be a finitely generated residually finite group and $\hat G$ its profinite completion. Then for all $g\in \hat G$ we have $gGg^{-1}\leq \hat G$ is dense. Suppose that $H\leq \hat G$ is a ...
4
votes
0answers
155 views

finite approximation equation on free group

An equation in a free group $F$ is an identity of the form $W(x_1,..,x_n,a_1,...a_n)=1$ where $x_1,...,x_n$ are variables, $a_1,...,a_n$ free generators of $F$ and $W$ a word in the free group on $x_1,...
0
votes
0answers
36 views

Is there a better rank bound for fibered products?

Let $G$ be a profinite group of rank $d \geq 2015$, which is a fibered product of two groups of rank at most $2$. That is, there exist closed normal subgroups $N_1, N_2 \lhd_c G$ with $N_1 \cap N_2 = \...
0
votes
1answer
166 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 ...
2
votes
1answer
197 views

Finite generation and profinite completion

Let $G$ be a (countable) residually finite group whose profinite completion is topologically finitely generated. Must $G$ be finitely generated?
5
votes
0answers
197 views

What is the kernel of the map $Out(\widehat{F_2})\rightarrow GL_2(\widehat{\mathbb{Z}})$?

Let $F_2$ be the free group on two generators, and $\widehat{F_2}$ its profinite completion. Let $Out(\widehat{F_2})$ be the outer automorphism group of $\widehat{F_2}$, ie, $Out(\widehat{F_2}) = Aut(\...
2
votes
0answers
109 views

Pro-G_p*G_q topology, profinite topology

Let $W=G_p*G_q$, where $G_p$ and $G_q$ are pseudovarieties of all finite $p$-groups and all finite $q$-groups respectively, with $p$ and $q$ fixed prime numbers. The pro-$W$ topology on a group $G$ is ...
2
votes
1answer
189 views

pseudovarieties and profinite group : do * and g() commute?

Let $V$ and $W$ be pseudovarieties of finite monoids. We denote with $gV$ the pseudovariety of categories generated by $V$, and by $V*W$ the semidirect product of pseudovarieties $V$ and $W$. Does ...
2
votes
0answers
109 views

Profinite topology

For a distance fixed prime numbers $p$ and $q$, let $G_p$, $G_q$ and $Sl$ the pseudovarieties of all finite $p$-group, $q$-group and semilattices respectively. Dose the following equality hold? $Sl*...
2
votes
0answers
188 views

Profinite Topology

Let $V$ and $W$ be pseudovarieties of finite groups. For a finite inverse monoid $M$, the $V$-kernel of $M$ is defined to be the intersection of all sets $f^{−1}(1)$, $f$ is a relational morphism ...
2
votes
0answers
154 views

Pro-p topology on free group

Let $H$ be a finitely generated subgroup of the free group $F(A)$ and $G_P$ the pseudovariety of all finite $p$-group with $p$ fixed prime number. We endow $F(A)$ with the pro-$G_p$ topology. Suppose ...
17
votes
0answers
307 views

Nonabelian topological fundamental group of a conjugate variety

Let $X$ be a pointed algebraic variety over the field of complex numbers $\mathbb{C}$. Let $\pi_1^{\rm top}(X)$ and $\pi_1^{\mathrm{\acute{e}t}}(X)$ denote the topological and the étale fundamental ...
3
votes
2answers
291 views

Adeles and twisted adeles

Let $\mu_n$ denote the group of $n$-th roots of unity in ${\mathbb{C}}$, i.e., $\mu_n=\ker[{\mathbb{C}}^*\overset{n}{\longrightarrow}{\mathbb{C}}^*]$. We set $$ \mu=\varinjlim_n \mu_n\subset {\mathbb{...
4
votes
0answers
103 views

Improvements of the Reidemeister-Schreier index formula for particular classes of groups

I have a couple of questions regarding possible improvements of the Reidemeister-Schreier index formula: let $G$ be a $d$-generated group and let $H$ be a subgroup of $G$, then $$d(H) \le (d-1) \...
14
votes
0answers
313 views

Is the absolute Galois group of the rationals Hopfian?

Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?